1,1,171,0,0.1187885,"\int (b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[(b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^2 B \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 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{2 B \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 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^2 B \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 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{2 B \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}",1,"(-6*b^3*B*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (6*b^2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*b*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*B*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",8,5,23,0.2174,1,"{3787, 3768, 3771, 2641, 2639}"
2,1,136,0,0.1009304,"\int (b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(-2*A*b^2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*A*b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*B*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",7,5,23,0.2174,1,"{3787, 3768, 3771, 2639, 2641}"
3,1,104,0,0.0822444,"\int \sqrt{b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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)}}","\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*b*B*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/d + (2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,5,23,0.2174,1,"{3787, 3771, 2641, 3768, 2639}"
4,1,82,0,0.0678043,"\int \frac{A+B \sec (c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/Sqrt[b*Sec[c + d*x]],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}","\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])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(b*d)","A",5,4,23,0.1739,1,"{3787, 3771, 2639, 2641}"
5,1,116,0,0.0913051,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(3/2),x]","\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)}}","\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,"(2*B*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*A*Sin[c + d*x])/(3*b*d*Sqrt[b*Sec[c + d*x]])","A",6,5,23,0.2174,1,"{3787, 3769, 3771, 2641, 2639}"
6,1,147,0,0.1041497,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(5/2),x]","\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 \sin (c+d x)}{3 b^2 d \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)}}{3 b^3 d}","\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 \sin (c+d x)}{3 b^2 d \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)}}{3 b^3 d}",1,"(6*A*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*A*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Sec[c + d*x]])","A",7,5,23,0.2174,1,"{3787, 3769, 3771, 2639, 2641}"
7,1,119,0,0.0982565,"\int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-4/3, 1/2, -1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(8/3)*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
8,1,116,0,0.0984483,"\int \sec (c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
9,1,112,0,0.0862797,"\int (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 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)}}","\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*A*b*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2])","A",5,3,23,0.1304,1,"{3787, 3772, 2643}"
10,1,115,0,0.1042793,"\int \cos (c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","-\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)}}","-\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*A*b^2*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
11,1,119,0,0.1211129,"\int \cos ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),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}}","-\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*A*b^3*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) - (3*b^2*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
12,1,119,0,0.101489,"\int \sec ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(10/3)*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
13,1,116,0,0.0985424,"\int \sec (c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
14,1,112,0,0.0880377,"\int (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*b*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])","A",5,3,23,0.1304,1,"{3787, 3772, 2643}"
15,1,115,0,0.1046781,"\int \cos (c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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}}","\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*A*b^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
16,1,119,0,0.1232232,"\int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","-\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}}","-\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*A*b^3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*b^2*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
17,1,117,0,0.0969972,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\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)}}","\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*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
18,1,114,0,0.0931026,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\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}}","\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*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
19,1,114,0,0.0865656,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{2/3}} \, dx","Int[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(2/3),x]","-\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}}","-\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*A*b*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",5,3,23,0.1304,1,"{3787, 3772, 2643}"
20,1,114,0,0.0933538,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\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}}","\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*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
21,1,117,0,0.0977646,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\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)}}","\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*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
22,1,117,0,0.096767,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 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)}}","\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*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
23,1,114,0,0.0931761,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),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)}}","-\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*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
24,1,114,0,0.0890296,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(4/3),x]","-\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}}","-\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*A*b*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",5,3,23,0.1304,1,"{3787, 3772, 2643}"
25,1,114,0,0.0931583,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),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)}}","-\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*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
26,1,117,0,0.0971938,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 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)}}","\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*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{16, 3787, 3772, 2643}"
27,1,167,0,0.1201745,"\int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*b*Hypergeometric2F1[1/2, (-1 - 3*m)/6, (5 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[1/2, (-4 - 3*m)/6, (2 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
28,1,165,0,0.1171121,"\int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-2);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right)}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}-\frac{3 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)}}","\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*A*Hypergeometric2F1[1/2, (1 - 3*m)/6, (7 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (-2 - 3*m)/6, (4 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(2 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
29,1,165,0,0.1117235,"\int \sec ^m(c+d x) \sqrt[3]{b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]),x]","\frac{3 B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-1);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right)}{d (3 m+1) \sqrt{\sin ^2(c+d x)}}-\frac{3 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)}}","\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*A*Hypergeometric2F1[1/2, (2 - 3*m)/6, (8 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (-1 - 3*m)/6, (5 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
30,1,165,0,0.1094311,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(1/3),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)}}","-\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*A*Hypergeometric2F1[1/2, (4 - 3*m)/6, (10 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1 - 3*m)/6, (7 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
31,1,165,0,0.1150007,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","-\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}}","-\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*A*Hypergeometric2F1[1/2, (5 - 3*m)/6, (11 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (2 - 3*m)/6, (8 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(2 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
32,1,173,0,0.1196103,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","-\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)}}","-\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*A*Hypergeometric2F1[1/2, (7 - 3*m)/6, (13 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (4 - 3*m)/6, (10 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(b*d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
33,1,172,0,0.1107611,"\int \sec ^m(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n);\frac{1}{2} (-m-n+2);\cos ^2(c+d x)\right)}{d (m+n) \sqrt{\sin ^2(c+d x)}}-\frac{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)}}","\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,"-((A*Hypergeometric2F1[1/2, (1 - m - n)/2, (3 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, (-m - n)/2, (2 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{20, 3787, 3772, 2643}"
34,1,143,0,0.1265532,"\int \sec ^2(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\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)}}","\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,"(A*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (-2 - n)/2, -n/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
35,1,136,0,0.1150485,"\int \sec (c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\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)}}","\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,"(A*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])","A",6,4,27,0.1481,1,"{16, 3787, 3772, 2643}"
36,1,137,0,0.0962599,"\int (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}-\frac{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)}}","\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,"-((A*b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])","A",5,3,21,0.1429,1,"{3787, 3772, 2643}"
37,1,151,0,0.1267329,"\int \cos (c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","-\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)}}","-\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,"-((A*b^2*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])) - (b*B*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])","A",6,4,27,0.1481,1,"{16, 3787, 3772, 2643}"
38,1,153,0,0.1421493,"\int \cos ^2(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","-\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)}}","-\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,"-((A*b^3*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*B*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{16, 3787, 3772, 2643}"
39,1,163,0,0.1164458,"\int \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\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)}}","\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*A*Hypergeometric2F1[1/2, (-1 - 2*n)/4, (3 - 2*n)/4, Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (-3 - 2*n)/4, (1 - 2*n)/4, Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
40,1,163,0,0.111319,"\int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),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)}}","\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*A*Hypergeometric2F1[1/2, (1 - 2*n)/4, (5 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (-1 - 2*n)/4, (3 - 2*n)/4, Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
41,1,163,0,0.1125925,"\int \frac{(b \sec (c+d x))^n (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],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)}}","-\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*A*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1 - 2*n)/4, (5 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
42,1,163,0,0.1128616,"\int \frac{(b \sec (c+d x))^n (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),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)}","-\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*A*Hypergeometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])","A",6,4,31,0.1290,1,"{20, 3787, 3772, 2643}"
43,1,134,0,0.1410874,"\int \sec ^4(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(3*a*(A + B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 4*B)*Tan[c + d*x])/(5*d) + (3*a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*B*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 4*B)*Tan[c + d*x]^3)/(15*d)","A",7,5,29,0.1724,1,"{3997, 3787, 3767, 3768, 3770}"
44,1,106,0,0.1234687,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(A + B)*Tan[c + d*x])/d + (a*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(A + B)*Tan[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3997, 3787, 3768, 3770, 3767}"
45,1,86,0,0.1154438,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 2*B)*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{3997, 3787, 3767, 8, 3768, 3770}"
46,1,56,0,0.0673097,"\int \sec (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(2*A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,27,0.1852,1,"{3997, 3787, 3770, 3767, 8}"
47,1,32,0,0.0331723,"\int (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{a B \tan (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 + B)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",4,4,21,0.1905,1,"{3914, 3767, 8, 3770}"
48,1,32,0,0.0473172,"\int \cos (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}","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 + B)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d","A",3,2,27,0.07407,1,"{3996, 3770}"
49,1,47,0,0.0864057,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(A + 2*B)*x)/2 + (a*(A + B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,29,0.1379,1,"{3996, 3787, 2637, 8}"
50,1,77,0,0.1083096,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(A + B)*x)/2 + (a*(2*A + 3*B)*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,29,0.1724,1,"{3996, 3787, 2635, 8, 2637}"
51,1,97,0,0.1187851,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(3*A + 4*B)*x)/8 + (a*(A + B)*Sin[c + d*x])/d + (a*(3*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3996, 3787, 2633, 2635, 8}"
52,1,125,0,0.1335275,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(3*a*(A + B)*x)/8 + (a*(4*A + 5*B)*Sin[c + d*x])/(5*d) + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*B)*Sin[c + d*x]^3)/(15*d)","A",7,5,29,0.1724,1,"{3996, 3787, 2635, 8, 2633}"
53,1,169,0,0.2444647,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(a^2*(7*A + 6*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(10*A + 9*B)*Tan[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*A + 6*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(5*d) + (a^2*(10*A + 9*B)*Tan[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4018, 3997, 3787, 3768, 3770, 3767}"
54,1,138,0,0.2284722,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(a^2*(8*A + 7*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*A + 7*B)*Tan[c + d*x])/(6*d) + (a^2*(8*A + 7*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A - B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)","A",7,7,31,0.2258,1,"{4010, 4001, 3788, 3767, 8, 4046, 3770}"
55,1,103,0,0.1133481,"\int \sec (c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*(3*A + 2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(3*A + 2*B)*Tan[c + d*x])/(3*d) + (a^2*(3*A + 2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{4001, 3788, 3767, 8, 4046, 3770}"
56,1,82,0,0.0836368,"\int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*A*x + (a^2*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B)*Tan[c + d*x])/(2*d) + (B*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",5,5,23,0.2174,1,"{3917, 3914, 3767, 8, 3770}"
57,1,73,0,0.1295213,"\int \cos (c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*(2*A + B)*x + (a^2*(A + 2*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A - B)*Sin[c + d*x])/d + (B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/d","A",4,3,29,0.1034,1,"{4018, 3996, 3770}"
58,1,88,0,0.1448741,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*(3*A + 4*B)*x)/2 + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A + 2*B)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",4,3,31,0.09677,1,"{4017, 3996, 3770}"
59,1,102,0,0.153287,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*(2*A + 3*B)*x)/2 + (2*a^2*(2*A + 3*B)*Sin[c + d*x])/(3*d) + (a^2*(2*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{4013, 3788, 2637, 4045, 8}"
60,1,135,0,0.2305165,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*(7*A + 8*B)*x)/8 + (a^2*(4*A + 5*B)*Sin[c + d*x])/(3*d) + (a^2*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*A + 4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(4*d)","A",6,6,31,0.1935,1,"{4017, 3996, 3787, 2635, 8, 2637}"
61,1,160,0,0.2515743,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(6*A + 7*B)*x)/8 + (a^2*(9*A + 10*B)*Sin[c + d*x])/(5*d) + (a^2*(6*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(6*A + 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(9*A + 10*B)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4017, 3996, 3787, 2633, 2635, 8}"
62,1,210,0,0.3993418,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(a^3*(26*A + 23*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(19*A + 17*B)*Tan[c + d*x])/(5*d) + (a^3*(26*A + 23*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(22*A + 21*B)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (a*B*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((3*A + 4*B)*Sec[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^3*(19*A + 17*B)*Tan[c + d*x]^3)/(15*d)","A",8,6,31,0.1935,1,"{4018, 3997, 3787, 3768, 3770, 3767}"
63,1,163,0,0.2680721,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(a^3*(15*A + 13*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(15*A + 13*B)*Tan[c + d*x])/(5*d) + (3*a^3*(15*A + 13*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*A - B)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (B*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(15*A + 13*B)*Tan[c + d*x]^3)/(60*d)","A",11,7,31,0.2258,1,"{4010, 4001, 3791, 3770, 3767, 8, 3768}"
64,1,125,0,0.1425578,"\int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(5*a^3*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(4*A + 3*B)*Tan[c + d*x])/d + (3*a^3*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (a^3*(4*A + 3*B)*Tan[c + d*x]^3)/(12*d)","A",10,6,29,0.2069,1,"{4001, 3791, 3770, 3767, 8, 3768}"
65,1,111,0,0.1442161,"\int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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^3*A*x + (a^3*(7*A + 5*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A + B)*Tan[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*A + 5*B)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",6,5,23,0.2174,1,"{3917, 3914, 3767, 8, 3770}"
66,1,108,0,0.2382154,"\int \cos (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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*(3*A + B)*x + (a^3*(6*A + 7*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*B*Sin[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((A + 2*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/d","A",5,3,29,0.1034,1,"{4018, 3996, 3770}"
67,1,117,0,0.2639387,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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*(7*A + 6*B)*x)/2 + (a^3*(A + 3*B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*A*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",5,4,31,0.1290,1,"{4017, 4018, 3996, 3770}"
68,1,125,0,0.2709568,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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*(5*A + 7*B)*x)/2 + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((5*A + 3*B)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",5,3,31,0.09677,1,"{4017, 3996, 3770}"
69,1,124,0,0.1695723,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(5*a^3*(3*A + 4*B)*x)/8 + (a^3*(3*A + 4*B)*Sin[c + d*x])/d + (3*a^3*(3*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(3*A + 4*B)*Sin[c + d*x]^3)/(12*d)","A",8,6,31,0.1935,1,"{4013, 3791, 2637, 2635, 8, 2633}"
70,1,176,0,0.3724173,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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*(13*A + 15*B)*x)/8 + (a^3*(38*A + 45*B)*Sin[c + d*x])/(15*d) + (a^3*(13*A + 15*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*(43*A + 45*B)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((7*A + 5*B)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(20*d)","A",7,6,31,0.1935,1,"{4017, 3996, 3787, 2635, 8, 2637}"
71,1,201,0,0.4097484,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(23*A + 26*B)*x)/16 + (a^3*(17*A + 19*B)*Sin[c + d*x])/(5*d) + (a^3*(23*A + 26*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(21*A + 22*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*A + 3*B)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(17*A + 19*B)*Sin[c + d*x]^3)/(15*d)","A",8,6,31,0.1935,1,"{4017, 3996, 3787, 2633, 2635, 8}"
72,1,194,0,0.3183613,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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}","\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,"(7*a^4*(8*A + 7*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(8*A + 7*B)*Tan[c + d*x])/(5*d) + (27*a^4*(8*A + 7*B)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(8*A + 7*B)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + ((6*A - B)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (B*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(8*A + 7*B)*Tan[c + d*x]^3)/(15*d)","A",14,7,31,0.2258,1,"{4010, 4001, 3791, 3770, 3767, 8, 3768}"
73,1,159,0,0.1792295,"\int \sec (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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}","\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,"(7*a^4*(5*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*(5*A + 4*B)*Tan[c + d*x])/(5*d) + (27*a^4*(5*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a^4*(5*A + 4*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + (4*a^4*(5*A + 4*B)*Tan[c + d*x]^3)/(15*d)","A",13,6,29,0.2069,1,"{4001, 3791, 3770, 3767, 8, 3768}"
74,1,151,0,0.2140043,"\int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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{(4 A+7 B) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 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{(4 A+7 B) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"a^4*A*x + (a^4*(48*A + 35*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*(8*A + 7*B)*Tan[c + d*x])/(8*d) + (a*B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + ((4*A + 7*B)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + ((32*A + 35*B)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)","A",7,5,23,0.2174,1,"{3917, 3914, 3767, 8, 3770}"
75,1,151,0,0.3679178,"\int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\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{(A+2 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 B \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}","-\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{(A+2 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 B \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"a^4*(4*A + B)*x + (a^4*(13*A + 12*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A + 2*B)*Sin[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((A + 2*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((9*A + 11*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",6,3,29,0.1034,1,"{4018, 3996, 3770}"
76,1,160,0,0.3895358,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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-B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}","\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-B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}",1,"(a^4*(13*A + 8*B)*x)/2 + (a^4*(8*A + 13*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(A - B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((A + 6*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",6,4,31,0.1290,1,"{4017, 4018, 3996, 3770}"
77,1,165,0,0.4098519,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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{(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{1}{2} a^4 x (12 A+13 B)+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}","\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{(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{1}{2} a^4 x (12 A+13 B)+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^4*(12*A + 13*B)*x)/2 + (a^4*(A + 4*B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(2*A + B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((2*A + B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((8*A - 3*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",6,4,31,0.1290,1,"{4017, 4018, 3996, 3770}"
78,1,173,0,0.4025829,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{5 a^4 (7 A+8 B) \sin (c+d x)}{8 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{(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{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}","\frac{5 a^4 (7 A+8 B) \sin (c+d x)}{8 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{(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{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^4*(35*A + 48*B)*x)/8 + (a^4*B*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 8*B)*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + ((7*A + 4*B)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + ((35*A + 32*B)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)","A",6,3,31,0.09677,1,"{4017, 3996, 3770}"
79,1,158,0,0.2017226,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(7*a^4*(4*A + 5*B)*x)/8 + (8*a^4*(4*A + 5*B)*Sin[c + d*x])/(5*d) + (27*a^4*(4*A + 5*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^4*(4*A + 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) - (4*a^4*(4*A + 5*B)*Sin[c + d*x]^3)/(15*d)","A",11,6,31,0.1935,1,"{4013, 3791, 2637, 2635, 8, 2633}"
80,1,220,0,0.5322535,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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{(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{(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{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 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{(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{(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{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(7*a^4*(7*A + 8*B)*x)/16 + (a^4*(72*A + 83*B)*Sin[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*A + 2*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + ((73*A + 72*B)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d)","A",8,6,31,0.1935,1,"{4017, 3996, 3787, 2635, 8, 2637}"
81,1,241,0,0.5669769,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\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{(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{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{a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 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{(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{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{a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}",1,"(a^4*(44*A + 49*B)*x)/16 + (a^4*(227*A + 252*B)*Sin[c + d*x])/(35*d) + (a^4*(44*A + 49*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(276*A + 301*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((10*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*d) + (7*(A + B)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(227*A + 252*B)*Sin[c + d*x]^3)/(105*d)","A",9,6,31,0.1935,1,"{4017, 3996, 3787, 2633, 2635, 8}"
82,1,131,0,0.1711128,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\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}","-\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)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((3*A - 4*B)*Tan[c + d*x])/(a*d) + (3*(A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A - 4*B)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,31,0.1613,1,"{4019, 3787, 3768, 3770, 3767}"
83,1,108,0,0.1627909,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-((2*A - 3*B)*ArcTanh[Sin[c + d*x]])/(2*a*d) + (2*(A - B)*Tan[c + d*x])/(a*d) - ((2*A - 3*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4019, 3787, 3767, 8, 3768, 3770}"
84,1,62,0,0.1167524,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"((A - B)*ArcTanh[Sin[c + d*x]])/(a*d) + (B*Tan[c + d*x])/(a*d) - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,31,0.1613,1,"{4008, 3787, 3770, 3767, 8}"
85,1,43,0,0.0818521,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}+\frac{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 \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(B*ArcTanh[Sin[c + d*x]])/(a*d) + ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",3,3,29,0.1034,1,"{3998, 3770, 3794}"
86,1,35,0,0.0591123,"\int \frac{A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]),x]","\frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(A*x)/a - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",2,2,23,0.08696,1,"{3919, 3794}"
87,1,60,0,0.1094684,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-(((A - B)*x)/a) + ((2*A - B)*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",4,4,29,0.1379,1,"{4020, 3787, 2637, 8}"
88,1,98,0,0.1498089,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"((3*A - 2*B)*x)/(2*a) - (2*(A - B)*Sin[c + d*x])/(a*d) + ((3*A - 2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,31,0.1613,1,"{4020, 3787, 2635, 8, 2637}"
89,1,122,0,0.1590809,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(-3*(A - B)*x)/(2*a) + ((4*A - 3*B)*Sin[c + d*x])/(a*d) - (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A - 3*B)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,31,0.1613,1,"{4020, 3787, 2633, 2635, 8}"
90,1,179,0,0.3211154,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((7*A - 10*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(2*A - 3*B)*Tan[c + d*x])/(a^2*d) + ((7*A - 10*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((7*A - 10*B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(2*A - 3*B)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,5,31,0.1613,1,"{4019, 3787, 3768, 3770, 3767}"
91,1,156,0,0.3055331,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"-((4*A - 7*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + (2*(5*A - 8*B)*Tan[c + d*x])/(3*a^2*d) - ((4*A - 7*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((5*A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,31,0.1935,1,"{4019, 3787, 3767, 8, 3768, 3770}"
92,1,108,0,0.2572173,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((A - 2*B)*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((A - 4*B)*Tan[c + d*x])/(3*a^2*d) - ((A - 2*B)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,31,0.1935,1,"{4019, 4008, 3787, 3770, 3767, 8}"
93,1,79,0,0.1872424,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(B*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*A - 5*B)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",4,4,31,0.1290,1,"{4008, 3998, 3770, 3794}"
94,1,65,0,0.0797186,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((A + 2*B)*Tan[c + d*x])/(3*d*(a^2 + a^2*Sec[c + d*x]))","A",2,2,29,0.06897,1,"{4000, 3794}"
95,1,70,0,0.1124705,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(A*x)/a^2 - ((4*A - B)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",3,3,23,0.1304,1,"{3922, 3919, 3794}"
96,1,98,0,0.2304901,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"-(((2*A - B)*x)/a^2) + (2*(5*A - 2*B)*Sin[c + d*x])/(3*a^2*d) - ((2*A - B)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",5,4,29,0.1379,1,"{4020, 3787, 2637, 8}"
97,1,143,0,0.3004007,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((7*A - 4*B)*x)/(2*a^2) - (2*(8*A - 5*B)*Sin[c + d*x])/(3*a^2*d) + ((7*A - 4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*A - 5*B)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,5,31,0.1613,1,"{4020, 3787, 2635, 8, 2637}"
98,1,170,0,0.3194364,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"-((10*A - 7*B)*x)/(2*a^2) + (4*(3*A - 2*B)*Sin[c + d*x])/(a^2*d) - ((10*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*A - 7*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(3*A - 2*B)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,5,31,0.1613,1,"{4020, 3787, 2633, 2635, 8}"
99,1,202,0,0.4747764,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((6*A - 13*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) + (8*(9*A - 19*B)*Tan[c + d*x])/(15*a^3*d) - ((6*A - 13*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*A - 11*B)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (4*(9*A - 19*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,31,0.1935,1,"{4019, 3787, 3767, 8, 3768, 3770}"
100,1,156,0,0.4285025,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((A - 3*B)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((7*A - 27*B)*Tan[c + d*x])/(15*a^3*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*A - 9*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((A - 3*B)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",7,6,31,0.1935,1,"{4019, 4008, 3787, 3770, 3767, 8}"
101,1,125,0,0.3153559,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(B*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((2*A - 7*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((4*A - 29*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",5,5,31,0.1613,1,"{4019, 4008, 3998, 3770, 3794}"
102,1,102,0,0.2031189,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A - 8*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + 7*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,31,0.09677,1,"{4008, 4000, 3794}"
103,1,102,0,0.1143322,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 3*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,29,0.1034,1,"{4000, 3796, 3794}"
104,1,108,0,0.1860893,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"(A*x)/a^3 - ((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 2*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (2*(11*A - B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,3,23,0.1304,1,"{3922, 3919, 3794}"
105,1,136,0,0.3667473,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-(((3*A - B)*x)/a^3) + (2*(36*A - 11*B)*Sin[c + d*x])/(15*a^3*d) - ((A - B)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - 4*B)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",6,4,29,0.1379,1,"{4020, 3787, 2637, 8}"
106,1,187,0,0.4695057,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((13*A - 6*B)*x)/(2*a^3) - (8*(19*A - 9*B)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (4*(19*A - 9*B)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",7,5,31,0.1613,1,"{4020, 3787, 2635, 8, 2637}"
107,1,218,0,0.4949202,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"-((23*A - 13*B)*x)/(2*a^3) + (4*(34*A - 19*B)*Sin[c + d*x])/(5*a^3*d) - ((23*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A - 13*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A - 19*B)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,5,31,0.1613,1,"{4020, 3787, 2633, 2635, 8}"
108,1,238,0,0.6555042,"\int \frac{\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^6*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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*A - 21*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + (8*(83*A - 216*B)*Tan[c + d*x])/(105*a^4*d) - ((8*A - 21*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) + ((52*A - 129*B)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*(83*A - 216*B)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",9,6,31,0.1935,1,"{4019, 3787, 3767, 8, 3768, 3770}"
109,1,194,0,0.6162409,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"((A - 4*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 244*B)*Tan[c + d*x])/(105*a^4*d) + ((25*A - 88*B)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((A - 4*B)*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((5*A - 12*B)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",8,6,31,0.1935,1,"{4019, 4008, 3787, 3770, 3767, 8}"
110,1,163,0,0.4751181,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"(B*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((6*A - 55*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((12*A - 215*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((3*A - 10*B)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",6,5,31,0.1613,1,"{4019, 4008, 3998, 3770, 3794}"
111,1,146,0,0.2286624,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"-((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((4*A + 3*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(4*A + 3*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((4*A + 3*B)*Tan[c + d*x])/(15*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,31,0.1290,1,"{4012, 3799, 4000, 3794}"
112,1,138,0,0.2646377,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"-((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((4*A - 11*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((8*A + 13*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((8*A + 13*B)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,31,0.1290,1,"{4008, 4000, 3796, 3794}"
113,1,138,0,0.1506584,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((3*A + 4*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (2*(3*A + 4*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*(3*A + 4*B)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,3,29,0.1034,1,"{4000, 3796, 3794}"
114,1,138,0,0.2678091,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"(A*x)/a^4 - ((55*A - 6*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (2*(80*A - 3*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((10*A - 3*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",5,3,23,0.1304,1,"{3922, 3919, 3794}"
115,1,166,0,0.5725927,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"-(((4*A - B)*x)/a^4) + (8*(83*A - 20*B)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",7,4,29,0.1379,1,"{4020, 3787, 2637, 8}"
116,1,223,0,0.6489948,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"((21*A - 8*B)*x)/(2*a^4) - (8*(216*A - 83*B)*Sin[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A - 52*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(216*A - 83*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",8,5,31,0.1613,1,"{4020, 3787, 2635, 8, 2637}"
117,1,256,0,0.7054548,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"-((44*A - 21*B)*x)/(2*a^4) + (8*(227*A - 108*B)*Sin[c + d*x])/(35*a^4*d) - ((44*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((178*A - 87*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((44*A - 21*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((16*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(227*A - 108*B)*Sin[c + d*x]^3)/(105*a^4*d)","A",9,5,31,0.1613,1,"{4020, 3787, 2633, 2635, 8}"
118,1,187,0,0.3381257,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(4*a*(9*A + 8*B)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*A + 8*B)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(9*A + 8*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (4*(9*A + 8*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)","A",5,5,33,0.1515,1,"{4016, 3803, 3800, 4001, 3792}"
119,1,144,0,0.2766736,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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*(7*A + 6*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(7*A + 6*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A + 6*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)","A",4,4,33,0.1212,1,"{4016, 3800, 4001, 3792}"
120,1,101,0,0.2276867,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}","\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*a*(5*A + 7*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)","A",3,3,33,0.09091,1,"{4010, 4001, 3792}"
121,1,62,0,0.0944036,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}","\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*a*(3*A + B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",2,2,31,0.06452,1,"{4001, 3792}"
122,1,66,0,0.0882574,"\int \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{a} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a B \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 \sqrt{a} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a B \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*B*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3915, 3774, 203, 3792}"
123,1,68,0,0.1063842,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a} (A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a A \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\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[a]*(A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,31,0.09677,1,"{4015, 3774, 203}"
124,1,117,0,0.1770774,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),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}}","\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,"(Sqrt[a]*(3*A + 4*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(3*A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,33,0.1212,1,"{4015, 3805, 3774, 203}"
125,1,160,0,0.242449,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(Sqrt[a]*(5*A + 6*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 6*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(5*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,33,0.1212,1,"{4015, 3805, 3774, 203}"
126,1,203,0,0.2980967,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(5*Sqrt[a]*(7*A + 8*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (5*a*(7*A + 8*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (5*a*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(7*A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])","A",6,4,33,0.1212,1,"{4015, 3805, 3774, 203}"
127,1,189,0,0.4610408,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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*(39*A + 34*B)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(9*A + 10*B)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(39*A + 34*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*B*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(39*A + 34*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)","A",5,5,33,0.1515,1,"{4018, 4016, 3800, 4001, 3792}"
128,1,138,0,0.2972706,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(8*a^2*(21*A + 19*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(21*A + 19*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A - 2*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)","A",4,4,33,0.1212,1,"{4010, 4001, 3793, 3792}"
129,1,101,0,0.1400355,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(8*a^2*(5*A + 3*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",3,3,31,0.09677,1,"{4001, 3793, 3792}"
130,1,105,0,0.1462753,"\int (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 (3 A+4 B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 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^{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 B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(3*A + 4*B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{3917, 3915, 3774, 203, 3792}"
131,1,103,0,0.2413941,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 (A-2 B) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (3 A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}","\frac{a^2 (A-2 B) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (3 A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(a^(3/2)*(3*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(A - 2*B)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,31,0.1290,1,"{4018, 4015, 3774, 203}"
132,1,119,0,0.2729872,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}","\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a^(3/2)*(7*A + 12*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,33,0.1212,1,"{4017, 4015, 3774, 203}"
133,1,164,0,0.365126,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 (11 A+14 B) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (11 A+14 B) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(11*A + 14*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(11*A + 14*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,33,0.1515,1,"{4017, 4015, 3805, 3774, 203}"
134,1,209,0,0.4486736,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 (75 A+88 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (75 A+88 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(75*A + 88*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(75*A + 88*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(9*A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,33,0.1515,1,"{4017, 4015, 3805, 3774, 203}"
135,1,237,0,0.6569264,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^2 (11 A+14 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}-\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}","\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^2 (11 A+14 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}-\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,"(2*a^3*(803*A + 710*B)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(209*A + 194*B)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(803*A + 710*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a^2*(11*A + 14*B)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*a*(803*A + 710*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*a*B*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",6,5,33,0.1515,1,"{4018, 4016, 3800, 4001, 3792}"
136,1,175,0,0.3527221,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{16 a^2 (15 A+13 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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}","\frac{16 a^2 (15 A+13 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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,"(64*a^3*(15*A + 13*B)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(15*A + 13*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(15*A + 13*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*A - 2*B)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*B*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)","A",5,4,33,0.1212,1,"{4010, 4001, 3793, 3792}"
137,1,138,0,0.1839632,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(64*a^3*(7*A + 5*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*A + 5*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",4,3,31,0.09677,1,"{4001, 3793, 3792}"
138,1,142,0,0.2227142,"\int (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 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^{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 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(35*A + 32*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(5*A + 8*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",6,5,25,0.2000,1,"{3917, 3915, 3774, 203, 3792}"
139,1,143,0,0.4105148,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","-\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{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{2 a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 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{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{2 a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(5*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*(3*A + 14*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(A + 2*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,4,31,0.1290,1,"{4018, 4015, 3774, 203}"
140,1,154,0,0.4187699,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^{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 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 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^{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 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^(5/2)*(19*A + 20*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(9*A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 4*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,5,33,0.1515,1,"{4017, 4018, 4015, 3774, 203}"
141,1,164,0,0.4564702,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^3 (49 A+54 B) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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^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}","\frac{a^3 (49 A+54 B) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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^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,"(a^(5/2)*(25*A + 38*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(3*A + 2*B)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,4,33,0.1212,1,"{4017, 4015, 3774, 203}"
142,1,209,0,0.580095,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^3 (163 A+200 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (95 A+104 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) (a \sec (c+d x)+a)^{3/2}}{4 d}","\frac{a^3 (163 A+200 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (95 A+104 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) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^(5/2)*(163*A + 200*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(163*A + 200*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(95*A + 104*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(11*A + 8*B)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,5,33,0.1515,1,"{4017, 4015, 3805, 3774, 203}"
143,1,254,0,0.653945,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^(5/2)*(283*A + 326*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(283*A + 326*B)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(157*A + 170*B)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,5,33,0.1515,1,"{4017, 4015, 3805, 3774, 203}"
144,1,202,0,0.6056219,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*A - 37*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(7*A - 31*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)","A",6,5,33,0.1515,1,"{4021, 4010, 4001, 3795, 203}"
145,1,159,0,0.4196555,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"(Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(5*A - 7*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*A - B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)","A",5,5,33,0.1515,1,"{4021, 4010, 4001, 3795, 203}"
146,1,118,0,0.2567717,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}","-\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,"-((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*A - 2*B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)","A",4,4,33,0.1212,1,"{4010, 4001, 3795, 203}"
147,1,78,0,0.1074474,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*B*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,31,0.09677,1,"{4001, 3795, 203}"
148,1,91,0,0.1071781,"\int \frac{A+B \sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/Sqrt[a + a*Sec[c + d*x]],x]","\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}","\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*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,4,25,0.1600,1,"{3920, 3774, 203, 3795}"
149,1,119,0,0.2294638,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"-(((A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,31,0.1613,1,"{4022, 3920, 3774, 203, 3795}"
150,1,165,0,0.3687988,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,5,33,0.1515,1,"{4022, 3920, 3774, 203, 3795}"
151,1,206,0,0.5545671,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((9*A - 14*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",8,5,33,0.1515,1,"{4022, 3920, 3774, 203, 3795}"
152,1,216,0,0.6329075,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((11*A - 15*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((65*A - 93*B)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*A - 9*B)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((35*A - 39*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)","A",6,6,33,0.1818,1,"{4019, 4021, 4010, 4001, 3795, 203}"
153,1,171,0,0.4610613,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((7*A - 11*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 13*B)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((3*A - 7*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)","A",5,5,33,0.1515,1,"{4019, 4010, 4001, 3795, 203}"
154,1,118,0,0.2588816,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((3*A - 7*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*B*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,33,0.1212,1,"{4008, 4001, 3795, 203}"
155,1,87,0,0.1217539,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((A + 3*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,31,0.09677,1,"{4000, 3795, 203}"
156,1,127,0,0.1814142,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,5,25,0.2000,1,"{3922, 3920, 3774, 203, 3795}"
157,1,170,0,0.4053042,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-(((3*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,31,0.1935,1,"{4020, 4022, 3920, 3774, 203, 3795}"
158,1,221,0,0.5832249,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((19*A - 12*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A - 6*B)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,33,0.1818,1,"{4020, 4022, 3920, 3774, 203, 3795}"
159,1,268,0,0.7796609,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((47*A - 38*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d) + ((17*A - 13*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (7*(3*A - 2*B)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A - 12*B)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,33,0.1818,1,"{4020, 4022, 3920, 3774, 203, 3795}"
160,1,216,0,0.6548866,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"-((75*A - 163*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((9*A - 17*B)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((93*A - 197*B)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 95*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)","A",6,5,33,0.1515,1,"{4019, 4010, 4001, 3795, 203}"
161,1,169,0,0.4549572,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((19*A - 75*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((A - 9*B)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,33,0.1515,1,"{4019, 4008, 4001, 3795, 203}"
162,1,126,0,0.2759044,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((5*A + 19*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 13*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,33,0.1212,1,"{4008, 4000, 3795, 203}"
163,1,126,0,0.1646411,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((3*A + 5*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*A + 5*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,31,0.1290,1,"{4000, 3796, 3795, 203}"
164,1,164,0,0.2537152,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 3*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,5,25,0.2000,1,"{3922, 3920, 3774, 203, 3795}"
165,1,207,0,0.5577517,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{(35 A-11 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\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{(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}}","\frac{(35 A-11 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\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{(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,"-(((5*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A - 11*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,31,0.1935,1,"{4020, 4022, 3920, 3774, 203, 3795}"
166,1,264,0,0.7903379,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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}}","-\frac{7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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,"((39*A - 20*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A - 11*B)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (7*(9*A - 5*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A - 15*B)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,33,0.1818,1,"{4020, 4022, 3920, 3774, 203, 3795}"
167,1,89,0,0.1464483,"\int \frac{A+A \sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[(A + A*Sec[c + d*x])/Sqrt[a - a*Sec[c + d*x]],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}","\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,4,26,0.1538,1,"{3904, 3887, 481, 203}"
168,1,115,0,0.2208909,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],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}","\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,"(3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])","A",6,5,32,0.1562,1,"{4022, 3920, 3774, 203, 3795}"
169,1,155,0,0.3619758,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],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)}}","\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,"(11*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (5*A*Sin[c + d*x])/(4*d*Sqrt[a - a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a - a*Sec[c + d*x]])","A",7,5,34,0.1471,1,"{4022, 3920, 3774, 203, 3795}"
170,1,192,0,0.5242769,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],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)}}","\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,"(23*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (9*A*Sin[c + d*x])/(8*d*Sqrt[a - a*Sec[c + d*x]]) + (7*A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a - a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a - a*Sec[c + d*x]])","A",8,5,34,0.1471,1,"{4022, 3920, 3774, 203, 3795}"
171,1,133,0,0.1995275,"\int \frac{A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx","Int[(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2),x]","\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 \sin (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{2 a d \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)}{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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(3/2)*d) - (3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) + (A*Csc[(c + d*x)/2]^2*Sin[c + d*x])/(2*a*d*Sqrt[a - a*Sec[c + d*x]])","A",6,5,26,0.1923,1,"{3904, 3887, 471, 522, 203}"
172,1,146,0,0.3547061,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(5*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(3/2)*d) - (7*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (2*A*Sin[c + d*x])/(a*d*Sqrt[a - a*Sec[c + d*x]])","A",7,6,32,0.1875,1,"{4020, 4022, 3920, 3774, 203, 3795}"
173,1,194,0,0.5291449,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(31*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (11*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (13*A*Sin[c + d*x])/(4*a*d*Sqrt[a - a*Sec[c + d*x]]) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a - a*Sec[c + d*x]])","A",8,6,34,0.1765,1,"{4020, 4022, 3920, 3774, 203, 3795}"
174,1,236,0,0.7011158,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","\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)}}","\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,"(85*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(3/2)*d) - (15*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (35*A*Sin[c + d*x])/(8*a*d*Sqrt[a - a*Sec[c + d*x]]) + (25*A*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a - a*Sec[c + d*x]]) + (4*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a - a*Sec[c + d*x]])","A",9,6,34,0.1765,1,"{4020, 4022, 3920, 3774, 203, 3795}"
175,1,185,0,0.2060168,"\int \frac{A+A \sec (c+d x)}{(a-a \sec (c+d x))^{5/2}} \, dx","Int[(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2),x]","\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 \sin (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{16 a^2 d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x) \cos (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d \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)}{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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(5/2)*d) - (23*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) + (7*A*Csc[(c + d*x)/2]^2*Sin[c + d*x])/(16*a^2*d*Sqrt[a - a*Sec[c + d*x]]) - (A*Cos[c + d*x]*Csc[(c + d*x)/2]^4*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])","A",7,6,26,0.2308,1,"{3904, 3887, 471, 527, 522, 203}"
176,1,184,0,0.5053325,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","\frac{23 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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}}","\frac{23 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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,"(7*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(5/2)*d) - (79*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (11*A*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (23*A*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])","A",8,6,32,0.1875,1,"{4020, 4022, 3920, 3774, 203, 3795}"
177,1,236,0,0.7308907,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","\frac{49 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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}}","\frac{49 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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,"(59*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(5/2)*d) - (167*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (15*A*Cos[c + d*x]*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (49*A*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (23*A*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])","A",9,6,34,0.1765,1,"{4020, 4022, 3920, 3774, 203, 3795}"
178,1,280,0,0.9062908,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","\frac{21 A \sin (c+d x)}{2 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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}}","\frac{21 A \sin (c+d x)}{2 a^2 d \sqrt{a-a \sec (c+d x)}}+\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{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,"(203*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(5/2)*d) - (287*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (19*A*Cos[c + d*x]^2*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (21*A*Sin[c + d*x])/(2*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (119*A*Cos[c + d*x]*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (77*A*Cos[c + d*x]^2*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x]])","A",10,6,34,0.1765,1,"{4020, 4022, 3920, 3774, 203, 3795}"
179,1,199,0,0.1793545,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (6*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*B*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,6,31,0.1935,1,"{3997, 3787, 3768, 3771, 2641, 2639}"
180,1,172,0,0.1645117,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-2*a*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,6,31,0.1935,1,"{3997, 3787, 3768, 3771, 2639, 2641}"
181,1,135,0,0.14369,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,31,0.1935,1,"{3997, 3787, 3771, 2641, 3768, 2639}"
182,1,106,0,0.1341137,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],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)}{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}","\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*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,5,31,0.1613,1,"{3997, 3787, 3771, 2639, 2641}"
183,1,110,0,0.1295598,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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)}}","\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,"(2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,31,0.1613,1,"{3996, 3787, 3771, 2639, 2641}"
184,1,141,0,0.1532392,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}","\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,"(2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,31,0.1935,1,"{3996, 3787, 3769, 3771, 2641, 2639}"
185,1,172,0,0.1613072,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,6,31,0.1935,1,"{3996, 3787, 3769, 3771, 2639, 2641}"
186,1,234,0,0.3420831,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*(7*A + 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*B*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d)","A",9,7,33,0.2121,1,"{4018, 3997, 3787, 3768, 3771, 2639, 2641}"
187,1,199,0,0.2993927,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-4*a^2*(5*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*(5*A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(5*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*B*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4018, 3997, 3787, 3771, 2641, 3768, 2639}"
188,1,160,0,0.2781658,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(-4*a^2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(3*A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",7,6,33,0.1818,1,"{4018, 3997, 3787, 3771, 2639, 2641}"
189,1,158,0,0.2557649,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","-\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}","-\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,"(4*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,33,0.1818,1,"{4017, 3997, 3787, 3771, 2639, 2641}"
190,1,166,0,0.2600258,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}","\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,"(4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(7*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",7,6,33,0.1818,1,"{4017, 3996, 3787, 3771, 2639, 2641}"
191,1,201,0,0.2898758,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(4*a^2*(3*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(9*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (4*a^2*(6*A + 7*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",8,7,33,0.2121,1,"{4017, 3996, 3787, 3769, 3771, 2641, 2639}"
192,1,234,0,0.3204089,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(4*a^2*(8*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(11*A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (4*a^2*(8*A + 9*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 6*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,7,33,0.2121,1,"{4017, 3996, 3787, 3769, 3771, 2639, 2641}"
193,1,277,0,0.5408784,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(21*A + 17*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(13*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^3*(24*A + 23*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*A + 13*B)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d)","A",10,7,33,0.2121,1,"{4018, 3997, 3787, 3768, 3771, 2639, 2641}"
194,1,244,0,0.4387354,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-4*a^3*(9*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(9*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(42*A + 41*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*A + 11*B)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d)","A",9,7,33,0.2121,1,"{4018, 3997, 3787, 3771, 2641, 3768, 2639}"
195,1,211,0,0.4159082,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(-4*a^3*(5*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(20*A + 21*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(5*A + 9*B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",8,6,33,0.1818,1,"{4018, 3997, 3787, 3771, 2639, 2641}"
196,1,199,0,0.4094202,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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)}}","\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,"(4*a^3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{4017, 4018, 3997, 3787, 3771, 2639, 2641}"
197,1,211,0,0.4130326,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","-\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)}","-\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,"(4*a^3*(9*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(6*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(9*A + 5*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",8,6,33,0.1818,1,"{4017, 3997, 3787, 3771, 2639, 2641}"
198,1,211,0,0.4370651,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(41*A + 42*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(11*A + 7*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",8,6,33,0.1818,1,"{4017, 3996, 3787, 3771, 2639, 2641}"
199,1,244,0,0.4766643,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(4*a^3*(17*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(23*A + 24*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^3*(11*A + 13*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(13*A + 9*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",9,7,33,0.2121,1,"{4017, 3996, 3787, 3769, 3771, 2641, 2639}"
200,1,277,0,0.5104607,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\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)}","\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,"(4*a^3*(15*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(21*A + 22*B)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^3*(15*A + 17*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 121*B)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(15*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))","A",10,7,33,0.2121,1,"{4017, 3996, 3787, 3769, 3771, 2639, 2641}"
201,1,229,0,0.2476698,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"(3*(5*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*(5*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) + (5*(A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((5*A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{4019, 3787, 3768, 3771, 2641, 2639}"
202,1,192,0,0.2271285,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"(-3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((3*A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,6,33,0.1818,1,"{4019, 3787, 3768, 3771, 2639, 2641}"
203,1,153,0,0.1862289,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"((A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{4019, 3787, 3771, 2641, 3768, 2639}"
204,1,123,0,0.1708146,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-(((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,5,33,0.1515,1,"{4019, 3787, 3771, 2639, 2641}"
205,1,128,0,0.1775179,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\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}","-\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,"((3*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,5,33,0.1515,1,"{4020, 3787, 3771, 2639, 2641}"
206,1,164,0,0.1942415,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\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}","\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,"(-3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A - 3*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2641, 2639}"
207,1,197,0,0.2125389,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\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}","-\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,"(3*(7*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A - 5*B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*(A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",8,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2639, 2641}"
208,1,230,0,0.2301892,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])),x]","-\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}","-\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,"(-21*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(9*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) + ((9*A - 7*B)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (7*(A - B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(9*A - 7*B)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2641, 2639}"
209,1,237,0,0.3711343,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (5*(A - 2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + ((4*A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,6,33,0.1818,1,"{4019, 3787, 3768, 3771, 2639, 2641}"
210,1,204,0,0.3452412,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"((A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((2*A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,6,33,0.1818,1,"{4019, 3787, 3771, 2641, 3768, 2639}"
211,1,161,0,0.3026374,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,5,33,0.1515,1,"{4019, 3787, 3771, 2639, 2641}"
212,1,168,0,0.3096358,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"-((A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((2*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,33,0.1818,1,"{4019, 4020, 3787, 3771, 2639, 2641}"
213,1,177,0,0.3256143,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\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}","-\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*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,5,33,0.1515,1,"{4020, 3787, 3771, 2639, 2641}"
214,1,211,0,0.3567152,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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*A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (5*(2*A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (5*(2*A - B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A - 4*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",8,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2641, 2639}"
215,1,244,0,0.3815879,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\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}","-\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,"(7*(8*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (7*(8*A - 5*B)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A - 2*B)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",9,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2639, 2641}"
216,1,292,0,0.5595424,"\int \frac{\sec ^{\frac{9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(-7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(7*A - 17*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((13*A - 33*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))","A",10,6,33,0.1818,1,"{4019, 3787, 3768, 3771, 2639, 2641}"
217,1,261,0,0.5362574,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"((9*A - 49*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((9*A - 49*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A - 8*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A - 13*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{4019, 3787, 3771, 2641, 3768, 2639}"
218,1,220,0,0.4898295,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",8,5,33,0.1515,1,"{4019, 3787, 3771, 2639, 2641}"
219,1,216,0,0.4836092,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,33,0.1818,1,"{4019, 4020, 3787, 3771, 2639, 2641}"
220,1,222,0,0.4894481,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((9*A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A + 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,33,0.1818,1,"{4019, 4020, 3787, 3771, 2639, 2641}"
221,1,228,0,0.4985139,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\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}","-\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*A - 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,5,33,0.1515,1,"{4020, 3787, 3771, 2639, 2641}"
222,1,261,0,0.5530048,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"(-7*(17*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2641, 2639}"
223,1,294,0,0.5701889,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\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}","-\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,"(7*(33*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((21*A - 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (7*(33*A - 17*B)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((21*A - 11*B)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - ((12*A - 7*B)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (3*(21*A - 11*B)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))","A",10,6,33,0.1818,1,"{4020, 3787, 3769, 3771, 2639, 2641}"
224,1,176,0,0.2884287,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),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}}","\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]*(6*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(6*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,35,0.1143,1,"{4016, 3803, 3801, 215}"
225,1,131,0,0.2371359,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),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}}","\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,"(Sqrt[a]*(4*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(4*A + 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{4016, 3803, 3801, 215}"
226,1,78,0,0.1602502,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(Sqrt[a]*(2*A + B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,35,0.08571,1,"{4016, 3801, 215}"
227,1,76,0,0.157209,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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}","\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*Sqrt[a]*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,35,0.08571,1,"{4015, 3801, 215}"
228,1,82,0,0.1582572,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}","\frac{2 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*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",2,2,35,0.05714,1,"{4013, 3804}"
229,1,130,0,0.2221026,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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}}","\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,"(2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,35,0.08571,1,"{4015, 3805, 3804}"
230,1,175,0,0.2877169,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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}}","\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*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(6*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(6*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])","A",4,3,35,0.08571,1,"{4015, 3805, 3804}"
231,1,227,0,0.5470717,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 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^{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 B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(a^(3/2)*(88*A + 75*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(88*A + 75*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{4018, 4016, 3803, 3801, 215}"
232,1,180,0,0.4185284,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 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^{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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a^(3/2)*(14*A + 11*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(14*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(6*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{4018, 4016, 3803, 3801, 215}"
233,1,133,0,0.336036,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 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^{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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a^(3/2)*(12*A + 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(4*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,35,0.1143,1,"{4018, 4016, 3801, 215}"
234,1,124,0,0.3135309,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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^{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 B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{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^{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 B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}",1,"(a^(3/2)*(2*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(2*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,35,0.1143,1,"{4018, 4015, 3801, 215}"
235,1,125,0,0.3392149,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((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 (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^{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 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}","\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^{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 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a^(3/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",4,4,35,0.1143,1,"{4017, 4015, 3801, 215}"
236,1,131,0,0.256374,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}","\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,"(8*a^2*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",3,3,35,0.08571,1,"{4013, 3809, 3804}"
237,1,181,0,0.4379682,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((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 (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)}","\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*(8*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(52*A + 63*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",4,4,35,0.1143,1,"{4017, 4015, 3805, 3804}"
238,1,228,0,0.5079689,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((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 (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)}","\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*(10*A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(34*A + 39*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,4,35,0.1143,1,"{4017, 4015, 3805, 3804}"
239,1,274,0,0.6930936,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^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^{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 B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\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^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^{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 B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^(5/2)*(326*A + 283*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(326*A + 283*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(170*A + 157*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(10*A + 13*B)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,5,35,0.1429,1,"{4018, 4016, 3803, 3801, 215}"
240,1,227,0,0.5946101,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 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^{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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^(5/2)*(200*A + 163*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(200*A + 163*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(104*A + 95*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 11*B)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{4018, 4016, 3803, 3801, 215}"
241,1,180,0,0.5134319,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 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^{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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(38*A + 25*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(54*A + 49*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,4,35,0.1143,1,"{4018, 4016, 3801, 215}"
242,1,180,0,0.504055,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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^{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 B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 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^{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 B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^(5/2)*(20*A + 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(4*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(4*A + 7*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,4,35,0.1143,1,"{4018, 4015, 3801, 215}"
243,1,177,0,0.5054451,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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{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{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{a^3 (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{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{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^(5/2)*(2*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(14*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(2*A - 3*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4017, 4018, 4015, 3801, 215}"
244,1,172,0,0.4889312,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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^{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 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a^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^{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 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*a^(5/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(32*A + 35*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",5,4,35,0.1143,1,"{4017, 4015, 3801, 215}"
245,1,178,0,0.3165179,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(64*a^3*(5*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",4,3,35,0.08571,1,"{4013, 3809, 3804}"
246,1,228,0,0.6315081,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((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 (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{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{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 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a^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{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{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 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*(124*A + 135*B)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(292*A + 345*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,4,35,0.1143,1,"{4017, 4015, 3805, 3804}"
247,1,275,0,0.6994858,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[((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 (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{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{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 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}","\frac{2 a^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{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{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 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*(194*A + 209*B)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(710*A + 803*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,4,35,0.1143,1,"{4017, 4015, 3805, 3804}"
248,1,190,0,0.5729242,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((4*A - 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((4*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4021, 4023, 3808, 206, 3801, 215}"
249,1,141,0,0.3871282,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"((2*A - B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,35,0.1714,1,"{4021, 4023, 3808, 206, 3801, 215}"
250,1,100,0,0.2324602,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}","\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*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,5,35,0.1429,1,"{4023, 3808, 206, 3801, 215}"
251,1,99,0,0.1850573,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\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}","\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,"-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,35,0.08571,1,"{4013, 3808, 206}"
252,1,142,0,0.3311708,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}}","-\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,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{4022, 4013, 3808, 206}"
253,1,187,0,0.5069524,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\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}}","\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,"-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,35,0.1143,1,"{4022, 4013, 3808, 206}"
254,1,230,0,0.6891396,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}}","-\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,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A - 91*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])","A",6,4,35,0.1143,1,"{4022, 4013, 3808, 206}"
255,1,247,0,0.7826862,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"-((12*A - 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) + ((9*A - 13*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((6*A - 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,35,0.2000,1,"{4019, 4021, 4023, 3808, 206, 3801, 215}"
256,1,197,0,0.6005483,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"((2*A - 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4019, 4021, 4023, 3808, 206, 3801, 215}"
257,1,145,0,0.3948687,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,6,35,0.1714,1,"{4019, 4023, 3808, 206, 3801, 215}"
258,1,107,0,0.194627,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((3*A + B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,35,0.08571,1,"{4012, 3808, 206}"
259,1,156,0,0.3618098,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"-((7*A - 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{4020, 4013, 3808, 206}"
260,1,203,0,0.552987,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"((11*A - 7*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A - 3*B)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A - 15*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4020, 4022, 4013, 3808, 206}"
261,1,250,0,0.7344689,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"-((15*A - 11*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 5*B)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B)*Sin[c + d*x])/(30*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((147*A - 95*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{4020, 4022, 4013, 3808, 206}"
262,1,246,0,0.8200142,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","-\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{(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{(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}}","-\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{(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{(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,"((2*A - 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 15*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((11*A - 35*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,35,0.2000,1,"{4019, 4021, 4023, 3808, 206, 3801, 215}"
263,1,194,0,0.5876625,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"(2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*A - 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,6,35,0.1714,1,"{4019, 4023, 3808, 206, 3801, 215}"
264,1,156,0,0.2715529,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((5*A + 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,35,0.1143,1,"{4012, 3810, 3808, 206}"
265,1,203,0,0.572176,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{(9 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(5 A+3 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}}","\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,"((19*A + 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((9*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4019, 4020, 4013, 3808, 206}"
266,1,203,0,0.5711684,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\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{(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{(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}}","\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{(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{(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,"-((75*A - 19*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,35,0.1143,1,"{4020, 4013, 3808, 206}"
267,1,250,0,0.7610362,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\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{(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{(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}}","-\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{(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{(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,"((163*A - 75*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((95*A - 39*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A - 147*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{4020, 4022, 4013, 3808, 206}"
268,1,297,0,0.9560272,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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{(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{(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}}","\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{(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{(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,"-((283*A - 163*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A - 85*B)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A - 1495*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,5,35,0.1429,1,"{4020, 4022, 4013, 3808, 206}"
269,1,406,0,0.6316006,"\int (a+a \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 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}}}","\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*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*B*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,25,0.3600,1,"{3924, 3779, 3778, 136, 3828, 3827, 50, 63, 225}"
270,1,354,0,0.366077,"\int \frac{A+B \sec (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/3),x]","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3^{3/4} 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}}","\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,"(3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",8,8,25,0.3200,1,"{3924, 3779, 3778, 136, 3828, 3827, 63, 225}"
271,1,415,0,0.4332995,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(4/3),x]","-\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(-\frac{5}{6};\frac{1}{2},1;\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 a 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}}","-\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,"(3*B*Tan[c + d*x])/(5*a*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-5/6, 1/2, 1, 1/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,25,0.3600,1,"{3924, 3779, 3778, 136, 3828, 3827, 51, 63, 225}"
272,1,787,0,0.8391878,"\int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\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}}}","\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,"(3*a*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) - (15*(1 + Sqrt[3])*a*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*3^(1/4)*a*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,25,0.4400,1,"{3924, 3779, 3778, 136, 3828, 3827, 50, 63, 308, 225, 1881}"
273,1,739,0,0.7023906,"\int \sqrt[3]{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[(a + a*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{2} A \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 d \sqrt{1-\sec (c+d x)}}-\frac{3 \left(1+\sqrt{3}\right) 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}}}","\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,"(3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",10,10,25,0.4000,1,"{3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881}"
274,1,764,0,0.7334512,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(2/3),x]","-\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}}","-\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,"(3*B*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-1/6, 1/2, 1, 5/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) + (3*(1 + Sqrt[3])*B*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (3*2^(1/3)*3^(1/4)*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (3^(3/4)*(1 - Sqrt[3])*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,25,0.4400,1,"{3924, 3779, 3778, 136, 3828, 3827, 51, 63, 308, 225, 1881}"
275,1,197,0,0.3608222,"\int (c \sec (e+f x))^n (a+a \sec (e+f x))^m (A+B \sec (e+f x)) \, dx","Int[(c*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m*(A + B*Sec[e + f*x]),x]","-\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)}}","-\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,"-((B*AppellF1[n, 1/2, -1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(c*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]])) - ((A - B)*AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(c*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]])","A",7,4,33,0.1212,1,"{4023, 3828, 3827, 133}"
276,1,164,0,0.2551285,"\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\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)}","\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*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n) + ((B + A*n + B*n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, (-2*Sec[c + d*x])/(1 - Sec[c + d*x])]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x]))","A",4,4,35,0.1143,1,"{4013, 3828, 3825, 132}"
277,1,114,0,0.1451076,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"((4*a*A + 3*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((A*b + a*B)*Tan[c + d*x])/d + ((4*a*A + 3*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A*b + a*B)*Tan[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3997, 3787, 3768, 3770, 3767}"
278,1,93,0,0.1327094,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"((A*b + a*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*a*A + 2*b*B)*Tan[c + d*x])/(3*d) + ((A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{3997, 3787, 3767, 8, 3768, 3770}"
279,1,61,0,0.0775628,"\int \sec (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"((2*a*A + b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((A*b + a*B)*Tan[c + d*x])/d + (b*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,27,0.1852,1,"{3997, 3787, 3770, 3767, 8}"
280,1,35,0,0.0346277,"\int (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\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 + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*B*Tan[c + d*x])/d","A",4,4,21,0.1905,1,"{3914, 3767, 8, 3770}"
281,1,35,0,0.0546407,"\int \cos (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","x (a B+A b)+\frac{a A \sin (c+d x)}{d}+\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 + a*B)*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d","A",3,2,27,0.07407,1,"{3996, 3770}"
282,1,52,0,0.0959616,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"((a*A + 2*b*B)*x)/2 + ((A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,29,0.1379,1,"{3996, 3787, 2637, 8}"
283,1,84,0,0.1253517,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"((A*b + a*B)*x)/2 + ((2*a*A + 3*b*B)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,29,0.1724,1,"{3996, 3787, 2635, 8, 2637}"
284,1,105,0,0.1387411,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"((3*a*A + 4*b*B)*x)/8 + ((A*b + a*B)*Sin[c + d*x])/d + ((3*a*A + 4*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3996, 3787, 2633, 2635, 8}"
285,1,198,0,0.2909145,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"((4*a^2*A + 3*A*b^2 + 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*b^2*B + 5*a*(2*A*b + a*B))*Tan[c + d*x])/(5*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*(5*A*b + 6*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (b*B*Sec[c + d*x]^3*(a + b*Sec[c + d*x])*Tan[c + d*x])/(5*d) + ((4*b^2*B + 5*a*(2*A*b + a*B))*Tan[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4026, 4047, 3767, 4046, 3768, 3770}"
286,1,179,0,0.322393,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\left(4 a^2 A b+a^3 (-B)+8 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 b 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{(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}","\frac{\left(4 a^2 A b+a^3 (-B)+8 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 b 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{(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,"((8*a*A*b + 4*a^2*B + 3*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*Tan[c + d*x])/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A*b - a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)","A",7,7,31,0.2258,1,"{4010, 4002, 3997, 3787, 3770, 3767, 8}"
287,1,116,0,0.1798849,"\int \sec (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"((2*a^2*A + A*b^2 + 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a*A*b + a^2*B + b^2*B)*Tan[c + d*x])/(3*d) + (b*(3*A*b + 2*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{4002, 3997, 3787, 3770, 3767, 8}"
288,1,86,0,0.0807489,"\int (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[(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 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}","\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,"a^2*A*x + ((4*a*A*b + 2*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(2*A*b + 3*a*B)*Tan[c + d*x])/(2*d) + (b*B*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",5,4,23,0.1739,1,"{3918, 3770, 3767, 8}"
289,1,60,0,0.102278,"\int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[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)}{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}","\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)*x + (b*(A*b + 2*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*A*Sin[c + d*x])/d + (b^2*B*Tan[c + d*x])/d","A",5,4,29,0.1379,1,"{4024, 3770, 3767, 8}"
290,1,80,0,0.1739255,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"((a^2*A + 2*A*b^2 + 4*a*b*B)*x)/2 + (b^2*B*ArcTanh[Sin[c + d*x]])/d + (a*(2*A*b + a*B)*Sin[c + d*x])/d + (a^2*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,31,0.1613,1,"{4024, 4047, 8, 4045, 3770}"
291,1,107,0,0.215749,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"((2*a*A*b + a^2*B + 2*b^2*B)*x)/2 + ((2*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{4024, 4047, 2637, 4045, 8}"
292,1,136,0,0.2599488,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"((3*a^2*A + 4*A*b^2 + 8*a*b*B)*x)/8 + ((2*a*A*b + a^2*B + b^2*B)*Sin[c + d*x])/d + ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(2*A*b + a*B)*Sin[c + d*x]^3)/(3*d)","A",7,6,31,0.1935,1,"{4024, 4047, 2635, 8, 4044, 3013}"
293,1,180,0,0.268378,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"((6*a*A*b + 3*a^2*B + 4*b^2*B)*x)/8 + ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4024, 4047, 2633, 4045, 2635, 8}"
294,1,252,0,0.4793903,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\left(15 a^3 A b+52 a^2 b^2 B-3 a^4 B+60 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 b d}+\frac{\left(12 a^2 A b+4 a^3 B+9 a b^2 B+3 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 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(30 a^2 A b-6 a^3 B+71 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 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}","\frac{\left(15 a^3 A b+52 a^2 b^2 B-3 a^4 B+60 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 b d}+\frac{\left(12 a^2 A b+4 a^3 B+9 a b^2 B+3 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 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(30 a^2 A b-6 a^3 B+71 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 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,"((12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*Tan[c + d*x])/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*A*b - a*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (B*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)","A",8,7,31,0.2258,1,"{4010, 4002, 3997, 3787, 3770, 3767, 8}"
295,1,180,0,0.3331844,"\int \sec (c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\left(16 a^2 A b+3 a^3 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{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{(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}","\frac{\left(16 a^2 A b+3 a^3 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{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{(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,"((8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Tan[c + d*x])/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,6,29,0.2069,1,"{4002, 3997, 3787, 3770, 3767, 8}"
296,1,137,0,0.1899279,"\int (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),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(6 a^2 A b+2 a^3 B+3 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 A x+\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}","\frac{b \left(8 a^2 B+9 a A b+2 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(6 a^2 A b+2 a^3 B+3 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 A x+\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,"a^3*A*x + ((6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(9*a*A*b + 8*a^2*B + 2*b^2*B)*Tan[c + d*x])/(3*d) + (b^2*(3*A*b + 5*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*B*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,5,23,0.2174,1,"{3918, 4048, 3770, 3767, 8}"
297,1,131,0,0.2232049,"\int \cos (c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\frac{b \left(2 a^2 A-3 a b B-A b^2\right) \tan (c+d x)}{d}+\frac{b \left(6 a^2 B+6 a A b+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (a B+3 A b)-\frac{b^2 (2 a A-b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \sin (c+d x) (a+b \sec (c+d x))^2}{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,"a^2*(3*A*b + a*B)*x + (b*(6*a*A*b + 6*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a^2*A - A*b^2 - 3*a*b*B)*Tan[c + d*x])/d - (b^2*(2*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,29,0.1724,1,"{4025, 4048, 3770, 3767, 8}"
298,1,124,0,0.3334507,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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}","\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,"(a*(a^2*A + 6*A*b^2 + 6*a*b*B)*x)/2 + (b^2*(A*b + 3*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a*A - 2*b*B)*Tan[c + d*x])/(2*d)","A",6,6,31,0.1935,1,"{4025, 4076, 4047, 8, 4045, 3770}"
299,1,145,0,0.347466,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(3 a^2 A b+a^3 B+6 a b^2 B+2 A b^3\right)+\frac{a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\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}","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(3 a^2 A b+a^3 B+6 a b^2 B+2 A b^3\right)+\frac{a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\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,"((3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*x)/2 + (b^3*B*ArcTanh[Sin[c + d*x]])/d + (a*(2*a^2*A + 8*A*b^2 + 9*a*b*B)*Sin[c + d*x])/(3*d) + (a^2*(5*A*b + 3*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{4025, 4074, 4047, 8, 4045, 3770}"
300,1,179,0,0.4234308,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\left(6 a^2 A b+2 a^3 B+9 a b^2 B+3 A b^3\right) \sin (c+d x)}{3 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{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^2 (2 a B+3 A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}","\frac{\left(6 a^2 A b+2 a^3 B+9 a b^2 B+3 A b^3\right) \sin (c+d x)}{3 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{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^2 (2 a B+3 A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*x)/8 + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*d) + (a*(3*a^2*A + 10*A*b^2 + 12*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(3*A*b + 2*a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,31,0.1935,1,"{4025, 4074, 4047, 2637, 4045, 8}"
301,1,221,0,0.4941933,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\frac{a \left(4 a^2 A+15 a b B+12 A b^2\right) \sin ^3(c+d x)}{15 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(9 a^2 A b+3 a^3 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(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right)+\frac{a^2 (5 a B+7 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 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{\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(9 a^2 A b+3 a^3 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(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right)+\frac{a^2 (5 a B+7 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*x)/8 + ((4*a^3*A + 14*a*A*b^2 + 15*a^2*b*B + 5*b^3*B)*Sin[c + d*x])/(5*d) + ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(7*A*b + 5*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a*(4*a^2*A + 12*A*b^2 + 15*a*b*B)*Sin[c + d*x]^3)/(15*d)","A",8,7,31,0.2258,1,"{4025, 4074, 4047, 2635, 8, 4044, 3013}"
302,1,334,0,0.7110258,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{\left(224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right) \tan (c+d x)}{60 b d}+\frac{\left(32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 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(24 a^2 A b-4 a^3 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(48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right) \tan (c+d x) \sec (c+d x)}{240 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}","\frac{\left(224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right) \tan (c+d x)}{60 b d}+\frac{\left(32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 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(24 a^2 A b-4 a^3 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(48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right) \tan (c+d x) \sec (c+d x)}{240 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,"((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]])/(16*d) + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Tan[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (B*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)","A",9,7,31,0.2258,1,"{4010, 4002, 3997, 3787, 3770, 3767, 8}"
303,1,250,0,0.5198752,"\int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{\left(95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 d}+\frac{\left(24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 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(130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 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}","\frac{\left(95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 d}+\frac{\left(24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 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(130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 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,"((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]])/(8*d) + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*Tan[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (B*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)","A",8,6,29,0.2069,1,"{4002, 3997, 3787, 3770, 3767, 8}"
304,1,200,0,0.3273519,"\int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{b \left(34 a^2 A b+19 a^3 B+16 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(32 a^3 A b+24 a^2 b^2 B+8 a^4 B+16 a A b^3+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^4 A x+\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}","\frac{b \left(34 a^2 A b+19 a^3 B+16 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(32 a^3 A b+24 a^2 b^2 B+8 a^4 B+16 a A b^3+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^4 A x+\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,"a^4*A*x + ((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]])/(8*d) + (b*(34*a^2*A*b + 4*A*b^3 + 19*a^3*B + 16*a*b^2*B)*Tan[c + d*x])/(6*d) + (b^2*(32*a*A*b + 26*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b*B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,6,23,0.2609,1,"{3918, 4056, 4048, 3770, 3767, 8}"
305,1,195,0,0.3683023,"\int \cos (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\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(12 a^2 A b+8 a^3 B+4 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\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}+a^3 x (a B+4 A b)-\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}","-\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(12 a^2 A b+8 a^3 B+4 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\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}+a^3 x (a B+4 A b)-\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)*x + (b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d - (b*(6*a^3*A - 12*a*A*b^2 - 17*a^2*b*B - 2*b^3*B)*Tan[c + d*x])/(3*d) - (b^2*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*a*A - b*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,29,0.2069,1,"{4025, 4056, 4048, 3770, 3767, 8}"
306,1,209,0,0.4628452,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\frac{b \left(13 a^2 A b+4 a^3 B-8 a b^2 B-2 A b^3\right) \tan (c+d x)}{2 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{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}","-\frac{b \left(13 a^2 A b+4 a^3 B-8 a b^2 B-2 A b^3\right) \tan (c+d x)}{2 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{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,"(a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*x)/2 + (b^2*(8*a*A*b + 12*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (b*(13*a^2*A*b - 2*A*b^3 + 4*a^3*B - 8*a*b^2*B)*Tan[c + d*x])/(2*d) - (b^2*(6*a*A*b + 2*a^2*B - b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,31,0.1935,1,"{4025, 4094, 4048, 3770, 3767, 8}"
307,1,198,0,0.5908146,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\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(4 a^2 A b+a^3 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}","\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(4 a^2 A b+a^3 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,"(a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*x)/2 + (b^3*(A*b + 4*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*Sin[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^2*(8*a*A*b + 3*a^2*B - 6*b^2*B)*Tan[c + d*x])/(6*d)","A",7,7,31,0.2258,1,"{4025, 4094, 4076, 4047, 8, 4045, 3770}"
308,1,216,0,0.6094933,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a \left(16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right) \sin (c+d x)}{6 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{1}{8} x \left(24 a^2 A b^2+3 a^4 A+16 a^3 b B+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}","\frac{a \left(16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right) \sin (c+d x)}{6 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{1}{8} x \left(24 a^2 A b^2+3 a^4 A+16 a^3 b B+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,"((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*x)/8 + (b^4*B*ArcTanh[Sin[c + d*x]])/d + (a*(16*a^2*A*b + 19*A*b^3 + 4*a^3*B + 34*a*b^2*B)*Sin[c + d*x])/(6*d) + (a^2*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(7*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{4025, 4094, 4074, 4047, 8, 4045, 3770}"
309,1,258,0,0.6905097,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{\left(60 a^2 A b^2+8 a^4 A+40 a^3 b B+60 a b^3 B+15 A b^4\right) \sin (c+d x)}{15 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(60 a^2 A b+15 a^3 B+110 a b^2 B+56 A b^3\right) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} x \left(12 a^3 A b+24 a^2 b^2 B+3 a^4 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}","\frac{\left(60 a^2 A b^2+8 a^4 A+40 a^3 b B+60 a b^3 B+15 A b^4\right) \sin (c+d x)}{15 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(60 a^2 A b+15 a^3 B+110 a b^2 B+56 A b^3\right) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} x \left(12 a^3 A b+24 a^2 b^2 B+3 a^4 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,"((12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*x)/8 + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*Sin[c + d*x])/(15*d) + (a*(60*a^2*A*b + 56*A*b^3 + 15*a^3*B + 110*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^2*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (a*(8*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (a*A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)","A",7,7,31,0.2258,1,"{4025, 4094, 4074, 4047, 2637, 4045, 8}"
310,1,309,0,0.8199453,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\frac{a \left(16 a^2 A b+4 a^3 B+27 a b^2 B+13 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(48 a^3 A b+87 a^2 b^2 B+12 a^4 B+53 a A b^3+15 b^4 B\right) \sin (c+d x)}{15 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{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+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(36 a^2 A b^2+5 a^4 A+24 a^3 b B+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}","-\frac{a \left(16 a^2 A b+4 a^3 B+27 a b^2 B+13 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(48 a^3 A b+87 a^2 b^2 B+12 a^4 B+53 a A b^3+15 b^4 B\right) \sin (c+d x)}{15 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{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+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(36 a^2 A b^2+5 a^4 A+24 a^3 b B+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,"((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*x)/16 + ((48*a^3*A*b + 53*a*A*b^3 + 12*a^4*B + 87*a^2*b^2*B + 15*b^4*B)*Sin[c + d*x])/(15*d) + ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (a*(3*A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (a*A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - (a*(16*a^2*A*b + 13*A*b^3 + 4*a^3*B + 27*a*b^2*B)*Sin[c + d*x]^3)/(15*d)","A",9,8,31,0.2581,1,"{4025, 4094, 4074, 4047, 2635, 8, 4044, 3013}"
311,1,187,0,0.675946,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \tan (c+d x)}{3 b^3 d}+\frac{\left(2 a^2+b^2\right) (A b-a B) \tanh ^{-1}(\sin (c+d x))}{2 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{(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}","-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \tan (c+d x)}{3 b^3 d}+\frac{\left(2 a^2+b^2\right) (A b-a B) \tanh ^{-1}(\sin (c+d x))}{2 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{(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,"((2*a^2 + b^2)*(A*b - a*B)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*Tan[c + d*x])/(3*b^3*d) + ((A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (B*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)","A",8,8,31,0.2581,1,"{4033, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
312,1,143,0,0.3954955,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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{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{(A b-a B) \tan (c+d x)}{b^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 b d}","-\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{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{(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,"-((2*a*A*b - 2*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + (2*a^2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((A*b - a*B)*Tan[c + d*x])/(b^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",7,7,31,0.2258,1,"{4033, 4082, 3998, 3770, 3831, 2659, 208}"
313,1,98,0,0.2288012,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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}","\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,"((A*b - a*B)*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*a*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Tan[c + d*x])/(b*d)","A",7,7,31,0.2258,1,"{4010, 12, 3789, 3770, 3831, 2659, 208}"
314,1,76,0,0.1264988,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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}","\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,"(B*ArcTanh[Sin[c + d*x]])/(b*d) + (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",5,5,29,0.1724,1,"{3998, 3770, 3831, 2659, 208}"
315,1,67,0,0.099078,"\int \frac{A+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]),x]","\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}}","\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*x)/a - (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)","A",4,4,23,0.1739,1,"{3919, 3831, 2659, 208}"
316,1,90,0,0.1490731,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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}","\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)*x)/a^2) + (2*b*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)","A",5,5,29,0.1724,1,"{4034, 12, 3783, 2659, 208}"
317,1,134,0,0.4031336,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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{x \left(a^2 A-2 a b B+2 A b^2\right)}{2 a^3}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}","-\frac{2 b^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{x \left(a^2 A-2 a b B+2 A b^2\right)}{2 a^3}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"((a^2*A + 2*A*b^2 - 2*a*b*B)*x)/(2*a^3) - (2*b^2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",6,6,31,0.1935,1,"{4034, 4104, 3919, 3831, 2659, 208}"
318,1,178,0,0.6418791,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2 A-3 a b B+3 A b^2\right) \sin (c+d x)}{3 a^3 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{x \left(a^2+2 b^2\right) (A b-a B)}{2 a^4}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}","\frac{\left(2 a^2 A-3 a b B+3 A b^2\right) \sin (c+d x)}{3 a^3 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{x \left(a^2+2 b^2\right) (A b-a B)}{2 a^4}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"-((a^2 + 2*b^2)*(A*b - a*B)*x)/(2*a^4) + (2*b^3*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B)*Sin[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)","A",7,6,31,0.1935,1,"{4034, 4104, 3919, 3831, 2659, 208}"
319,1,240,0,0.9823212,"\int \frac{\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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{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{x \left(4 a^2 A b^2+3 a^4 A-4 a^3 b B-8 a b^3 B+8 A b^4\right)}{8 a^5}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}","-\frac{\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{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{x \left(4 a^2 A b^2+3 a^4 A-4 a^3 b B-8 a b^3 B+8 A b^4\right)}{8 a^5}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"((3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*x)/(8*a^5) - (2*b^4*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*a^2 + 3*b^2)*(A*b - a*B)*Sin[c + d*x])/(3*a^4*d) + ((3*a^2*A + 4*A*b^2 - 4*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,6,31,0.1935,1,"{4034, 4104, 3919, 3831, 2659, 208}"
320,1,272,0,0.8655904,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2 A b-3 a^3 B+2 a b^2 B-A b^3\right) \tan (c+d x)}{b^3 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{2 a^2 \left(2 a^2 A b-3 a^3 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}}+\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(2 a^2 A b-3 a^3 B+2 a b^2 B-A b^3\right) \tan (c+d x)}{b^3 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{2 a^2 \left(2 a^2 A b-3 a^3 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}}+\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)}",1,"-((4*a*A*b - 6*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,8,31,0.2581,1,"{4029, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
321,1,164,0,0.5781801,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","-\frac{2 a \left(a^2 A b-2 a^3 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^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{(A b-2 a B) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{B \tan (c+d x)}{b^2 d}","-\frac{2 a \left(a^2 A b-2 a^3 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^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{(A b-2 a B) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{B \tan (c+d x)}{b^2 d}",1,"((A*b - 2*a*B)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (B*Tan[c + d*x])/(b^2*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,31,0.2258,1,"{4028, 4082, 3998, 3770, 3831, 2659, 208}"
322,1,131,0,0.3005458,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","-\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}","-\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,"(B*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(A*b - a*B)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4009, 3998, 3770, 3831, 2659, 208}"
323,1,100,0,0.1344402,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\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))}","\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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((A*b - a*B)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,29,0.1724,1,"{4003, 12, 3831, 2659, 208}"
324,1,124,0,0.2074548,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2,x]","-\frac{2 \left(2 a^2 A b+a^3 (-B)-A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{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(2 a^2 A b+a^3 (-B)-A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{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}",1,"(A*x)/a^2 - (2*(2*a^2*A*b - A*b^3 - a^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,23,0.2174,1,"{3923, 3919, 3831, 2659, 208}"
325,1,180,0,0.5689461,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\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{2 b \left(3 a^2 A b-2 a^3 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}}+\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{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{2 b \left(3 a^2 A b-2 a^3 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}}+\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{x (2 A b-a B)}{a^3}",1,"-(((2*A*b - a*B)*x)/a^3) + (2*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2*A - 2*A*b^2 + a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,29,0.2069,1,"{4030, 4104, 3919, 3831, 2659, 208}"
326,1,261,0,0.8908389,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","-\frac{\left(2 a^2 A b+a^3 (-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{\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{2 b^2 \left(4 a^2 A b-3 a^3 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}}+\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(2 a^2 A b+a^3 (-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{\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{2 b^2 \left(4 a^2 A b-3 a^3 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}}+\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}",1,"((a^2*A + 6*A*b^2 - 4*a*b*B)*x)/(2*a^4) - (2*b^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,6,31,0.1935,1,"{4030, 4104, 3919, 3831, 2659, 208}"
327,1,346,0,1.2745581,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(7 a^2 A b^2+2 a^4 A-6 a^3 b B+9 a b^3 B-12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\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{\left(2 a^2 A b+a^3 (-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(5 a^2 A b-4 a^3 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{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{x \left(2 a^2 A b+a^3 (-B)-6 a b^2 B+8 A b^3\right)}{2 a^5}","\frac{\left(7 a^2 A b^2+2 a^4 A-6 a^3 b B+9 a b^3 B-12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\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{\left(2 a^2 A b+a^3 (-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(5 a^2 A b-4 a^3 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{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{x \left(2 a^2 A b+a^3 (-B)-6 a b^2 B+8 A b^3\right)}{2 a^5}",1,"-((2*a^2*A*b + 8*A*b^3 - a^3*B - 6*a*b^2*B)*x)/(2*a^5) + (2*b^3*(5*a^2*A*b - 4*A*b^3 - 4*a^3*B + 3*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((2*a^4*A + 7*a^2*A*b^2 - 12*A*b^4 - 6*a^3*b*B + 9*a*b^3*B)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) - ((2*a^2*A*b - 4*A*b^3 - a^3*B + 3*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) + ((a^2*A - 4*A*b^2 + 3*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,6,31,0.1935,1,"{4030, 4104, 3919, 3831, 2659, 208}"
328,1,407,0,1.959388,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-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{\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^2 \left(-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-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}}+\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{a \left(2 a^2 A b-4 a^3 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(3 a^3 A b+10 a^2 b^2 B-6 a^4 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(-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-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{\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^2 \left(-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-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}}+\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{a \left(2 a^2 A b-4 a^3 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(3 a^3 A b+10 a^2 b^2 B-6 a^4 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}",1,"-((6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",9,9,31,0.2903,1,"{4029, 4098, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
329,1,289,0,1.42402,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","-\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 \left(-5 a^2 A b^3+2 a^4 A b+15 a^3 b^2 B-6 a^5 B-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 (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{a^2 \left(a^2 A b-3 a^3 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 b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}","-\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 \left(-5 a^2 A b^3+2 a^4 A b+15 a^3 b^2 B-6 a^5 B-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 (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{a^2 \left(a^2 A b-3 a^3 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 b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((A*b - 3*a*B)*ArcTanh[Sin[c + d*x]])/(b^4*d) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,31,0.2581,1,"{4029, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
330,1,220,0,0.6864627,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(a^2 A b^3+5 a^3 b^2 B-2 a^5 B-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{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(a^2 A b-3 a^3 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{B \tanh ^{-1}(\sin (c+d x))}{b^3 d}","\frac{\left(a^2 A b^3+5 a^3 b^2 B-2 a^5 B-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{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(a^2 A b-3 a^3 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{B \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(B*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,31,0.2258,1,"{4028, 4080, 3998, 3770, 3831, 2659, 208}"
331,1,180,0,0.3355138,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","-\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{\left(a^2 A b+a^3 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))}+\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^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{\left(a^2 A b+a^3 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))}+\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}",1,"-(((3*a*A*b - a^2*B - 2*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(A*b - a*B)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4009, 4003, 12, 3831, 2659, 208}"
332,1,164,0,0.2641682,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\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}","\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*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b - a*B)*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,5,29,0.1724,1,"{4003, 12, 3831, 2659, 208}"
333,1,205,0,0.5363154,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(-5 a^2 A b^3+6 a^4 A b-a^3 b^2 B-2 a^5 B+2 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b \left(5 a^2 A b-3 a^3 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{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{A x}{a^3}","-\frac{\left(-5 a^2 A b^3+6 a^4 A b-a^3 b^2 B-2 a^5 B+2 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b \left(5 a^2 A b-3 a^3 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{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{A x}{a^3}",1,"(A*x)/a^3 - ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,23,0.2609,1,"{3923, 4060, 3919, 3831, 2659, 208}"
334,1,290,0,1.5349473,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(-11 a^2 A b^2+2 a^4 A+5 a^3 b B-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(-15 a^2 A b^3+12 a^4 A b+5 a^3 b^2 B-6 a^5 B-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}}+\frac{b \left(6 a^2 A b-4 a^3 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{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{x (3 A b-a B)}{a^4}","\frac{\left(-11 a^2 A b^2+2 a^4 A+5 a^3 b B-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(-15 a^2 A b^3+12 a^4 A b+5 a^3 b^2 B-6 a^5 B-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}}+\frac{b \left(6 a^2 A b-4 a^3 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{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{x (3 A b-a B)}{a^4}",1,"-(((3*A*b - a*B)*x)/a^4) + (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(6*a^2*A*b - 3*A*b^3 - 4*a^3*B + a*b^2*B)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,29,0.2414,1,"{4030, 4100, 4104, 3919, 3831, 2659, 208}"
335,1,393,0,1.9989258,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(-21 a^2 A b^3+6 a^4 A b+11 a^3 b^2 B-2 a^5 B-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{\left(-10 a^2 A b^2+a^4 A+6 a^3 b B-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{b^2 \left(-29 a^2 A b^3+20 a^4 A b+15 a^3 b^2 B-12 a^5 B-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}}+\frac{b \left(7 a^2 A b-5 a^3 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{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{\left(-21 a^2 A b^3+6 a^4 A b+11 a^3 b^2 B-2 a^5 B-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{\left(-10 a^2 A b^2+a^4 A+6 a^3 b B-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{b^2 \left(-29 a^2 A b^3+20 a^4 A b+15 a^3 b^2 B-12 a^5 B-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}}+\frac{b \left(7 a^2 A b-5 a^3 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{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}",1,"((a^2*A + 12*A*b^2 - 6*a*b*B)*x)/(2*a^5) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,7,31,0.2258,1,"{4030, 4100, 4104, 3919, 3831, 2659, 208}"
336,1,418,0,5.2734396,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(3 a^3 A b+23 a^2 b^2 B-12 a^4 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 \left(-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+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 (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(a^2 A b-4 a^3 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{a^2 \left(-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-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 b-4 a B) \tanh ^{-1}(\sin (c+d x))}{b^5 d}","-\frac{\left(3 a^3 A b+23 a^2 b^2 B-12 a^4 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 \left(-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+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 (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(a^2 A b-4 a^3 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{a^2 \left(-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-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 b-4 a B) \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"((A*b - 4*a*B)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((3*a^3*A*b - 8*a*A*b^3 - 12*a^4*B + 23*a^2*b^2*B - 6*b^4*B)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,9,31,0.2903,1,"{4029, 4098, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
337,1,310,0,1.3663888,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(3 a^2 A b^5-7 a^5 b^2 B+8 a^3 b^4 B+2 a^7 B-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{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(a^2 A b^3-28 a^3 b^2 B+9 a^5 B+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{B \tanh ^{-1}(\sin (c+d x))}{b^4 d}","-\frac{\left(3 a^2 A b^5-7 a^5 b^2 B+8 a^3 b^4 B+2 a^7 B-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{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(a^2 A b^3-28 a^3 b^2 B+9 a^5 B+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{B \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(B*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 34*a*b^4*B)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,8,31,0.2581,1,"{4029, 4090, 4080, 3998, 3770, 3831, 2659, 208}"
338,1,274,0,0.6998921,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\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^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{a \left(a^2 A b-4 a^3 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(-10 a^2 A b^3+a^4 A b-5 a^3 b^2 B+2 a^5 B+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))}","\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^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{a \left(a^2 A b-4 a^3 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(-10 a^2 A b^3+a^4 A b-5 a^3 b^2 B+2 a^5 B+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,"((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,7,31,0.2258,1,"{4028, 4080, 4003, 12, 3831, 2659, 208}"
339,1,263,0,0.6150226,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(4 a^2 A b+a^3 (-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(2 a^3 A b-10 a^2 b^2 B+a^4 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))}+\frac{\left(2 a^2 A b+a^3 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{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(4 a^2 A b+a^3 (-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(2 a^3 A b-10 a^2 b^2 B+a^4 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))}+\frac{\left(2 a^2 A b+a^3 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{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}",1,"-(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b - a*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,31,0.1935,1,"{4009, 4003, 12, 3831, 2659, 208}"
340,1,237,0,0.5099449,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\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(11 a^2 A b-2 a^3 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))}-\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(11 a^2 A b-2 a^3 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))}-\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}",1,"((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b - a*B)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,5,29,0.1724,1,"{4003, 12, 3831, 2659, 208}"
341,1,292,0,1.0676358,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left(-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+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{b \left(8 a^2 A b-5 a^3 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 (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{A x}{a^4}","-\frac{\left(-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left(-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+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{b \left(8 a^2 A b-5 a^3 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 (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{A x}{a^4}",1,"(A*x)/a^4 - ((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{3923, 4060, 3919, 3831, 2659, 208}"
342,1,411,0,5.5950616,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\left(-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+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(-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+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}}+\frac{b \left(-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-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{b \left(9 a^2 A b-6 a^3 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 (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{x (4 A b-a B)}{a^5}","\frac{\left(-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+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(-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+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}}+\frac{b \left(-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-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{b \left(9 a^2 A b-6 a^3 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 (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{x (4 A b-a B)}{a^5}",1,"-(((4*A*b - a*B)*x)/a^5) + (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,7,29,0.2414,1,"{4030, 4100, 4104, 3919, 3831, 2659, 208}"
343,1,538,0,6.8442773,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+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{b^2 \left(-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+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}}+\frac{b \left(-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-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{b \left(10 a^2 A b-7 a^3 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 (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{\left(-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+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{b^2 \left(-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+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}}+\frac{b \left(-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-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{b \left(10 a^2 A b-7 a^3 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 (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}",1,"((a^2*A + 20*A*b^2 - 8*a*b*B)*x)/(2*a^6) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,7,31,0.2258,1,"{4030, 4100, 4104, 3919, 3831, 2659, 208}"
344,1,61,0,0.1147404,"\int \frac{\frac{b B}{a}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Int[((b*B)/a + B*Sec[c + d*x])/(a + b*Sec[c + d*x]),x]","\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}","\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*x)/a^2 + (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*d)","A",4,4,28,0.1429,1,"{3919, 3831, 2659, 208}"
345,1,6,0,0.0013912,"\int \frac{\frac{a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Int[((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",2,2,28,0.07143,1,"{21, 8}"
346,1,86,0,0.1804828,"\int \frac{a+b \sec (c+d x)}{(b+a \sec (c+d x))^2} \, dx","Int[(a + b*Sec[c + d*x])/(b + a*Sec[c + d*x])^2,x]","-\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)}","-\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,"(a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(b^2*d) - (a*Tan[c + d*x])/(b*d*(b + a*Sec[c + d*x]))","A",5,5,23,0.2174,1,"{3923, 3919, 3831, 2659, 205}"
347,1,87,0,0.0737429,"\int \frac{3+\sec (c+d x)}{2-\sec (c+d x)} \, dx","Int[(3 + Sec[c + d*x])/(2 - Sec[c + d*x]),x]","-\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}","-\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,"(3*x)/2 - (5*Log[Cos[(c + d*x)/2] - Sqrt[3]*Sin[(c + d*x)/2]])/(2*Sqrt[3]*d) + (5*Log[Cos[(c + d*x)/2] + Sqrt[3]*Sin[(c + d*x)/2]])/(2*Sqrt[3]*d)","A",4,4,21,0.1905,1,"{3919, 3831, 2659, 207}"
348,1,485,0,1.4367143,"\int \sec ^4(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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(12 a^2 A b-8 a^3 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(12 a^2 b (2 A-B)-16 a^3 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(24 a^3 A b-24 a^2 b^2 B-16 a^4 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}","\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(12 a^2 A b-8 a^3 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(12 a^2 b (2 A-B)-16 a^3 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(24 a^3 A b-24 a^2 b^2 B-16 a^4 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,"(-2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) + 18*a*b^2*(A - 2*B) + 12*a^2*b*(2*A - B) - 16*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*A*b - 75*A*b^3 - 8*a^3*B - 13*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b + a*B)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*B*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)","A",7,7,33,0.2121,1,"{4031, 4102, 4092, 4082, 4005, 3832, 4004}"
349,1,397,0,0.9327894,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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(14 a^2 A b-8 a^3 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}","\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(14 a^2 A b-8 a^3 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,"(2*(a - b)*Sqrt[a + b]*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) + 2*a*b*(7*A - 3*B) - 8*a^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(7*a*A*b - 4*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*A*b + a*B)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b*d) + (2*B*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*d)","A",6,6,33,0.1818,1,"{4031, 4092, 4082, 4005, 3832, 4004}"
350,1,314,0,0.5975016,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(-2*(a - b)*Sqrt[a + b]*(5*a*A*b - 2*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*(a - b)*Sqrt[a + b]*(5*A*b - 2*a*B - 9*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*A*b - 2*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)","A",5,5,33,0.1515,1,"{4010, 4002, 4005, 3832, 4004}"
351,1,256,0,0.3401921,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(a - b)*Sqrt[a + b]*(3*A*b + a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*(a - b)*Sqrt[a + b]*(3*A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",4,4,31,0.1290,1,"{4002, 4005, 3832, 4004}"
352,1,320,0,0.2904962,"\int \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}","\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*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(A*b + (a - b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",5,5,25,0.2000,1,"{3916, 3784, 4005, 3832, 4004}"
353,1,344,0,0.3689397,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}","\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,"(A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A + 2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,31,0.1935,1,"{4032, 4058, 3921, 3784, 3832, 4004}"
354,1,429,0,0.7327712,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","-\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}","-\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 - b)*Sqrt[a + b]*(A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2*A - A*b^2 + 4*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,33,0.2121,1,"{4032, 4104, 4058, 3921, 3784, 3832, 4004}"
355,1,509,0,1.1294917,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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} \left(4 a^2 A b+8 a^3 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{\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{(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}","\frac{\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} \left(4 a^2 A b+8 a^3 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{\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{(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"((a - b)*Sqrt[a + b]*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) + (Sqrt[a + b]*(2*a + b)*(8*a*A - 3*A*b + 6*a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (Sqrt[a + b]*(4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{4032, 4104, 4058, 3921, 3784, 3832, 4004}"
356,1,475,0,1.2246162,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","-\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(18 a^2 A b-8 a^3 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(-6 a^2 b (3 A-B)+8 a^3 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(18 a^3 A b-33 a^2 b^2 B-8 a^4 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}","-\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(18 a^2 A b-8 a^3 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(-6 a^2 b (3 A-B)+8 a^3 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(18 a^3 A b-33 a^2 b^2 B-8 a^4 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,"(2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) - 3*a*b^2*(57*A - 13*B) - 6*a^2*b*(3*A - B) + 8*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*A*b - 8*a^2*B - 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)","A",7,6,33,0.1818,1,"{4033, 4082, 4002, 4005, 3832, 4004}"
357,1,388,0,0.8290646,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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(21 a^2 A b-6 a^3 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}","\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(21 a^2 A b-6 a^3 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,"(-2*(a - b)*Sqrt[a + b]*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) + 6*a^2*B - a*(21*A*b - 57*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(21*a*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*B*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)","A",6,5,33,0.1515,1,"{4010, 4002, 4005, 3832, 4004}"
358,1,312,0,0.5702945,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(a - b)*Sqrt[a + b]*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(a - b)*Sqrt[a + b]*(15*a*A - 5*A*b - 3*a*B + 9*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",5,4,31,0.1290,1,"{4002, 4005, 3832, 4004}"
359,1,381,0,0.464855,"\int (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","-\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}","-\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,"(-2*(a - b)*Sqrt[a + b]*(3*A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*Sqrt[a + b]*(b^2*(3*A - B) - 3*a^2*B - a*(6*A*b - 4*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,25,0.2400,1,"{3918, 4058, 3921, 3784, 3832, 4004}"
360,1,361,0,0.4514438,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(a*A - 2*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(2*b*(A - B) + a*(A + 4*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(3*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,31,0.1935,1,"{4025, 4058, 3921, 3784, 3832, 4004}"
361,1,428,0,0.7921637,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","-\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}","-\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 - b)*Sqrt[a + b]*(5*A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(2*a*A + 5*A*b + 4*a*B + 8*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*A + 3*A*b^2 + 12*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((5*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,33,0.2121,1,"{4025, 4104, 4058, 3921, 3784, 3832, 4004}"
362,1,520,0,1.2752829,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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(12 a^2 A b+8 a^3 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}","\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(12 a^2 A b+8 a^3 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,"((a - b)*Sqrt[a + b]*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*A + 14*a*A*b + 3*A*b^2 + 12*a^2*B + 30*a*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) - (Sqrt[a + b]*(12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((7*A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a*A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{4025, 4104, 4058, 3921, 3784, 3832, 4004}"
363,1,566,0,1.7843746,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","-\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(110 a^2 A b-40 a^3 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(110 a^3 A b-285 a^2 b^2 B-40 a^4 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(-15 a^2 b^2 (121 A-19 B)-a^3 (110 A b-30 b B)+40 a^4 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(-3069 a^2 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-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}","-\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(110 a^2 A b-40 a^3 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(110 a^3 A b-285 a^2 b^2 B-40 a^4 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(-15 a^2 b^2 (121 A-19 B)-a^3 (110 A b-30 b B)+40 a^4 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(-3069 a^2 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-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,"(2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(a - b)*Sqrt[a + b]*(6*a*b^3*(209*A - 505*B) - 3*b^4*(539*A - 225*B) - 15*a^2*b^2*(121*A - 19*B) + 40*a^4*B - a^3*(110*A*b - 30*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B - 81*b^2*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)","A",8,6,33,0.1818,1,"{4033, 4082, 4002, 4005, 3832, 4004}"
364,1,469,0,1.1828498,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(45 a^2 A b-10 a^3 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(15 a^2 b (3 A-11 B)-10 a^3 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(45 a^3 A b+279 a^2 b^2 B-10 a^4 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}","\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(45 a^2 A b-10 a^3 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(15 a^2 b (3 A-11 B)-10 a^3 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(45 a^3 A b+279 a^2 b^2 B-10 a^4 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,"(-2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) - 6*a*b^2*(60*A - 19*B) + 15*a^2*b*(3*A - 11*B) - 10*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*B*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)","A",7,5,33,0.1515,1,"{4010, 4002, 4005, 3832, 4004}"
365,1,384,0,0.8067953,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(161 a^2 A b+15 a^3 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}","\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(161 a^2 A b+15 a^3 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,"(-2*(a - b)*Sqrt[a + b]*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) - 8*a*b*(7*A - 15*B) + 15*a^2*(7*A - B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",6,4,31,0.1290,1,"{4002, 4005, 3832, 4004}"
366,1,442,0,0.6558753,"\int (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{a+b} \left(a^2 b (45 A-23 B)+15 a^3 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 (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 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}","\frac{2 \sqrt{a+b} \left(a^2 b (45 A-23 B)+15 a^3 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 (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 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,"(-2*(a - b)*Sqrt[a + b]*(35*a*A*b + 23*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*Sqrt[a + b]*(a^2*b*(45*A - 23*B) - a*b^2*(35*A - 17*B) + b^3*(5*A - 9*B) + 15*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*(5*A*b + 8*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,7,25,0.2800,1,"{3918, 4056, 4058, 3921, 3784, 3832, 4004}"
367,1,433,0,0.7032329,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(3*a^2*A - 6*A*b^2 - 14*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(2*a*b*(9*A - 7*B) - 2*b^2*(3*A - B) + 3*a^2*(A + 6*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (a*Sqrt[a + b]*(5*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*a*A - 2*b*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,31,0.2258,1,"{4025, 4056, 4058, 3921, 3784, 3832, 4004}"
368,1,450,0,0.8347809,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(9*a*A*b + 4*a^2*B - 8*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(8*b^2*(A - B) + 2*a^2*(A + 2*B) + 3*a*b*(3*A + 8*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*A + 15*A*b^2 + 20*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (a*(7*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",7,7,33,0.2121,1,"{4025, 4094, 4058, 3921, 3784, 3832, 4004}"
369,1,518,0,1.3651896,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(20 a^2 A b+8 a^3 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}","\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(20 a^2 A b+8 a^3 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,"((a - b)*Sqrt[a + b]*(16*a^2*A + 33*A*b^2 + 54*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2*A + 26*a*A*b + 33*A*b^2 + 12*a^2*B + 54*a*b*B + 48*b^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2*A + 33*A*b^2 + 54*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(3*A*b + 2*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,33,0.2424,1,"{4025, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
370,1,617,0,1.8322091,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\left(284 a^2 A b+128 a^3 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{\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{\sqrt{a+b} \left(4 a^2 b (71 A+52 B)+8 a^3 (9 A+16 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(284 a^2 A b+128 a^3 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(120 a^2 A b^2+48 a^4 A+160 a^3 b B+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}","\frac{\left(284 a^2 A b+128 a^3 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{\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{\sqrt{a+b} \left(4 a^2 b (71 A+52 B)+8 a^3 (9 A+16 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(284 a^2 A b+128 a^3 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(120 a^2 A b^2+48 a^4 A+160 a^3 b B+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,"((a - b)*Sqrt[a + b]*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B) + 4*a^2*b*(71*A + 52*B) + 2*a*b^2*(59*A + 132*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(11*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",9,8,33,0.2424,1,"{4025, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
371,1,329,0,0.6175192,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\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 (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 (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}","\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 (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 (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,"(2*(a - b)*Sqrt[a + b]*(10*a*A*b - 8*a^2*B - 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (2*Sqrt[a + b]*(b^2*(5*A - 9*B) - 8*a^2*B + 2*a*b*(5*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*B*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",5,5,33,0.1515,1,"{4033, 4082, 4005, 3832, 4004}"
372,1,261,0,0.3971097,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","-\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 (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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b 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 (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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(-2*(a - b)*Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) - (2*Sqrt[a + b]*(3*A*b - (2*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)","A",4,4,33,0.1212,1,"{4010, 4005, 3832, 4004}"
373,1,210,0,0.2059722,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\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}","\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*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)","A",3,3,31,0.09677,1,"{4005, 3832, 4004}"
374,1,208,0,0.1240494,"\int \frac{A+B \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","\frac{2 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,"(2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",3,3,25,0.1200,1,"{3921, 3784, 3832}"
375,1,348,0,0.404588,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b} (A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{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}","\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,"(A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (A*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b]*(A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",6,6,31,0.1935,1,"{4034, 4059, 3921, 3784, 3832, 4004}"
376,1,435,0,0.7238142,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","-\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{(3 A b-4 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}-\frac{\sqrt{a+b} (3 A b-2 a (A+2 B)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{(a-b) \sqrt{a+b} (3 A b-4 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 b d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}","-\frac{\sqrt{a+b} \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{(3 A b-4 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}-\frac{\sqrt{a+b} (3 A b-2 a (A+2 B)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{(a-b) \sqrt{a+b} (3 A b-4 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 b d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"-((a - b)*Sqrt[a + b]*(3*A*b - 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*b*d) - (Sqrt[a + b]*(3*A*b - 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(4*a^2*A + 3*A*b^2 - 4*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - ((3*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)","A",7,7,33,0.2121,1,"{4034, 4104, 4058, 3921, 3784, 3832, 4004}"
377,1,525,0,1.167635,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\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(4 a^2 A b-8 a^3 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{(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{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a 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(4 a^2 A b-8 a^3 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{(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{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"((a - b)*Sqrt[a + b]*(16*a^2*A + 15*A*b^2 - 18*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*b*d) + (Sqrt[a + b]*(16*a^2*A - 10*a*A*b + 15*A*b^2 + 12*a^2*B - 18*a*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*d) + (Sqrt[a + b]*(4*a^2*A*b + 5*A*b^3 - 8*a^3*B - 6*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^4*d) + ((16*a^2*A + 15*A*b^2 - 18*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^3*d) - ((5*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a^2*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",8,7,33,0.2121,1,"{4034, 4104, 4058, 3921, 3784, 3832, 4004}"
378,1,329,0,0.7196534,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","-\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(6 a^2 A b-8 a^3 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}","-\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(6 a^2 A b-8 a^3 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,"(-2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) - (2*(2*a + b)*(3*A*b - (4*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*(A*b - a*B)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)","A",5,5,33,0.1515,1,"{4028, 4082, 4005, 3832, 4004}"
379,1,275,0,0.4640354,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*(a*A*b - 2*a^2*B + b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b - (2*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(A*b - a*B)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,33,0.1212,1,"{4009, 4005, 3832, 4004}"
380,1,254,0,0.3481559,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(-2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*(A + B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(A*b - a*B)*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,31,0.1290,1,"{4003, 4005, 3832, 4004}"
381,1,376,0,0.4332591,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*(A*b - a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{3923, 4058, 3921, 3784, 3832, 4004}"
382,1,427,0,0.7005138,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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{\sqrt{a+b} (3 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}","\frac{b \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{\sqrt{a+b} (3 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((a^2*A - 3*A*b^2 + 2*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((3*A*b + a*(A - 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2*A - 3*A*b^2 + 2*a*b*B)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,31,0.2258,1,"{4034, 4061, 4058, 3921, 3784, 3832, 4004}"
383,1,531,0,1.1392532,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{b \left(7 a^2 A b-4 a^3 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(-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{\left(7 a^2 A b-4 a^3 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{\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{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}","-\frac{b \left(7 a^2 A b-4 a^3 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(-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{\left(7 a^2 A b-4 a^3 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{\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{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"-((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*b*Sqrt[a + b]*d) - ((15*A*b^2 + a*b*(5*A - 12*B) - 2*a^2*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(4*a^2*A + 15*A*b^2 - 12*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - ((5*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,33,0.2424,1,"{4034, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
384,1,630,0,1.6694901,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),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{b \left(41 a^2 A b^2+16 a^4 A-42 a^3 b B+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(-6 a^2 b (A+5 B)+4 a^3 (4 A+3 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{\left(41 a^2 A b^2+16 a^4 A-42 a^3 b B+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{\sqrt{a+b} \left(12 a^2 A b-8 a^3 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{(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{A \sin (c+d x) \cos ^2(c+d x)}{3 a 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{b \left(41 a^2 A b^2+16 a^4 A-42 a^3 b B+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(-6 a^2 b (A+5 B)+4 a^3 (4 A+3 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{\left(41 a^2 A b^2+16 a^4 A-42 a^3 b B+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{\sqrt{a+b} \left(12 a^2 A b-8 a^3 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{(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{A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}",1,"((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^4*b*Sqrt[a + b]*d) + ((105*A*b^3 + 5*a*b^2*(7*A - 18*B) + 4*a^3*(4*A + 3*B) - 6*a^2*b*(A + 5*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^4*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a^2*A*b + 35*A*b^3 - 8*a^3*B - 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^5*d) + ((16*a^2*A + 35*A*b^2 - 30*a*b*B)*Sin[c + d*x])/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((7*A*b - 6*a*B)*Cos[c + d*x]*Sin[c + d*x])/(12*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (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)*Tan[c + d*x])/(24*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,33,0.2424,1,"{4034, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
385,1,510,0,1.5881382,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\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 a^2 \left(3 a^2 A b-6 a^3 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(-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 \left(-2 a^2 b^2 (3 A+8 B)-a^3 (8 A b-12 b B)+16 a^4 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(-15 a^2 A b^3+8 a^4 A b+28 a^3 b^2 B-16 a^5 B-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}}","\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 a^2 \left(3 a^2 A b-6 a^3 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(-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 \left(-2 a^2 b^2 (3 A+8 B)-a^3 (8 A b-12 b B)+16 a^4 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(-15 a^2 A b^3+8 a^4 A b+28 a^3 b^2 B-16 a^5 B-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,"(-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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) + (2*(9*a*b^3*(A - B) + b^4*(3*A - B) + 16*a^4*B - 2*a^2*b^2*(3*A + 8*B) - a^3*(8*A*b - 12*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",6,6,33,0.1818,1,"{4029, 4090, 4082, 4005, 3832, 4004}"
386,1,417,0,1.0157269,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","-\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(2 a^2 A b-5 a^3 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(2 a^2 b (A-3 B)-8 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(2 a^3 A b+15 a^2 b^2 B-8 a^4 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}}","-\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(2 a^2 A b-5 a^3 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(2 a^2 b (A-3 B)-8 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(2 a^3 A b+15 a^2 b^2 B-8 a^4 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,"(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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) + (2*(2*a^2*b*(A - 3*B) - 3*b^3*(A - B) - 8*a^3*B + 3*a*b^2*(A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (2*a^2*(A*b - a*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,33,0.1515,1,"{4028, 4080, 4005, 3832, 4004}"
387,1,387,0,0.6936622,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(a^2 A b+2 a^3 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 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(a^2 A b+2 a^3 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}}","\frac{2 \left(a^2 A b+2 a^3 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 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(a^2 A b+2 a^3 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,"(2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*B - 3*b^2*(A + B) + a*b*(A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,33,0.1515,1,"{4009, 4003, 4005, 3832, 4004}"
388,1,353,0,0.5980994,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(-2*(4*a*A*b - a^2*B - 3*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(3*a*A - A*b + a*B - 3*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*(A*b - a*B)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,4,31,0.1290,1,"{4003, 4005, 3832, 4004}"
389,1,495,0,0.7669106,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 b \left(7 a^2 A b-4 a^3 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 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(6 a^2 A b+a^2 b B-3 a^3 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}}+\frac{2 \left(7 a^2 A b-4 a^3 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 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 \left(7 a^2 A b-4 a^3 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 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(6 a^2 A b+a^2 b B-3 a^3 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}}+\frac{2 \left(7 a^2 A b-4 a^3 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 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}",1,"(2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*(6*a^2*A*b - a*A*b^2 - 3*A*b^3 - 3*a^3*B + a^2*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3923, 4060, 4058, 3921, 3784, 3832, 4004}"
390,1,582,0,1.2142039,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{b \left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-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{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(-a^2 b (21 A+2 B)-3 a^3 (A-4 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{\left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-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{\sqrt{a+b} (5 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}","\frac{b \left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-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{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(-a^2 b (21 A+2 B)-3 a^3 (A-4 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{\left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-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{\sqrt{a+b} (5 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) - ((15*A*b^3 + a*b^2*(5*A - 6*B) - 3*a^3*(A - 4*B) - a^2*b*(21*A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (Sqrt[a + b]*(5*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^2*A - 5*A*b^2 + 2*a*b*B)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (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)*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,31,0.2581,1,"{4034, 4061, 4060, 4058, 3921, 3784, 3832, 4004}"
391,1,686,0,2.0504654,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{b \left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-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{b \left(27 a^2 A b-12 a^3 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(-5 a^2 b^2 (27 A+4 B)-a^3 (27 A b-84 b B)+6 a^4 (A+2 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{\left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-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{\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{(7 A b-4 a B) \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}","-\frac{b \left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-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{b \left(27 a^2 A b-12 a^3 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(-5 a^2 b^2 (27 A+4 B)-a^3 (27 A b-84 b B)+6 a^4 (A+2 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{\left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-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{\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{(7 A b-4 a B) \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"-((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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*b*(a + b)^(3/2)*d) + ((105*A*b^4 + 5*a*b^3*(7*A - 12*B) + 6*a^4*(A + 2*B) - 5*a^2*b^2*(27*A + 4*B) - a^3*(27*A*b - 84*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(4*a^2*A + 35*A*b^2 - 20*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - ((7*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (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)*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,33,0.2424,1,"{4034, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
392,1,105,0,0.0812832,"\int \frac{\sec (e+f x) (A+A \sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[(Sec[e + f*x]*(A + A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],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}","-\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,"(-2*A*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f)","A",1,1,31,0.03226,1,"{4004}"
393,1,107,0,0.0849392,"\int \frac{\sec (e+f x) (A-A \sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],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}","\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,"(2*A*Sqrt[a - b]*(a + b)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a - b]], (a - b)/(a + b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f)","A",1,1,32,0.03125,1,"{4004}"
394,1,180,0,0.1825514,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-2*(5*a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*a*A + 3*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(A*b + a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,6,31,0.1935,1,"{3997, 3787, 3768, 3771, 2639, 2641}"
395,1,143,0,0.1466295,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*b + a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,31,0.1935,1,"{3997, 3787, 3771, 2641, 3768, 2639}"
396,1,111,0,0.1388485,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],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)}{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}","\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*(a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*b*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,5,31,0.1613,1,"{3997, 3787, 3771, 2639, 2641}"
397,1,115,0,0.1472808,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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)}}","\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,"(2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,31,0.1613,1,"{3996, 3787, 3771, 2639, 2641}"
398,1,148,0,0.1634423,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}","\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,"(2*(3*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,31,0.1935,1,"{3996, 3787, 3769, 3771, 2641, 2639}"
399,1,180,0,0.1813517,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a*A + 7*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,6,31,0.1935,1,"{3996, 3787, 3769, 3771, 2639, 2641}"
400,1,263,0,0.37276,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(7*A*b + 9*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)","A",9,7,33,0.2121,1,"{4026, 4047, 3768, 3771, 2641, 4046, 2639}"
401,1,221,0,0.31499,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(-2*(3*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(5*A*b + 7*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4026, 4047, 3768, 3771, 2639, 4046, 2641}"
402,1,177,0,0.2704193,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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}","\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*(a^2*A - A*b^2 - 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(3*A*b + 5*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",7,6,33,0.1818,1,"{4026, 4047, 3771, 2641, 4046, 2639}"
403,1,161,0,0.2477616,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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}","\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,"(-2*(b^2*B - a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*b^2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,33,0.1818,1,"{4024, 4047, 3771, 2641, 4046, 2639}"
404,1,171,0,0.260082,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}}","\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,"(2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,33,0.1818,1,"{4024, 4047, 3771, 2639, 4045, 2641}"
405,1,213,0,0.2887221,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4024, 4047, 3769, 3771, 2641, 4045, 2639}"
406,1,254,0,0.3372045,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(2*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(7*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(7*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,33,0.2121,1,"{4024, 4047, 3769, 3771, 2639, 4045, 2641}"
407,1,345,0,0.572322,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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(21 a^2 A b+7 a^3 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(21 a^2 A b+7 a^3 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}","\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(21 a^2 A b+7 a^3 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(21 a^2 A b+7 a^3 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,"(-2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*A*b + 13*a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)","A",10,8,33,0.2424,1,"{4026, 4076, 4047, 3768, 3771, 2641, 4046, 2639}"
408,1,295,0,0.5044271,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\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(15 a^2 A b+5 a^3 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(15 a^2 A b+5 a^3 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}","\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(15 a^2 A b+5 a^3 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(15 a^2 A b+5 a^3 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*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)","A",9,8,33,0.2424,1,"{4026, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
409,1,244,0,0.4830273,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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(9 a^2 A b+3 a^3 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}","\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(9 a^2 A b+3 a^3 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,"(2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(15*a*A*b + 14*a^2*B + 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*(5*A*b + 9*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4026, 4076, 4047, 3771, 2641, 4046, 2639}"
410,1,239,0,0.5101851,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","-\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(3 a^2 A b+a^3 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)}}","-\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(3 a^2 A b+a^3 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,"(2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(2*a^2*A - 3*A*b^2 - 9*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(a*A - b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4025, 4076, 4047, 3771, 2641, 4046, 2639}"
411,1,236,0,0.4612407,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{2 \left(3 a^2 A b+a^3 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 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\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)}","\frac{2 \left(3 a^2 A b+a^3 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 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\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,"(2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(9*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(a*A - 5*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",8,7,33,0.2121,1,"{4025, 4074, 4047, 3771, 2641, 4046, 2639}"
412,1,245,0,0.4667282,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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 \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(9 a^2 A b+3 a^3 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^2 (7 a B+11 A b) \sin (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))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\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 \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(9 a^2 A b+3 a^3 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^2 (7 a B+11 A b) \sin (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))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(11*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a*(5*a^2*A + 18*A*b^2 + 21*a*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",8,7,33,0.2121,1,"{4025, 4074, 4047, 3771, 2639, 4045, 2641}"
413,1,295,0,0.5383181,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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 \left(15 a^2 A b+5 a^3 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(15 a^2 A b+5 a^3 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^2 (9 a B+13 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\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 \left(15 a^2 A b+5 a^3 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(15 a^2 A b+5 a^3 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^2 (9 a B+13 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\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,"(2*(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]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(13*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a*(7*a^2*A + 22*A*b^2 + 27*a*b*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,8,33,0.2424,1,"{4025, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
414,1,345,0,0.5739197,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\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 \left(21 a^2 A b+7 a^3 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(21 a^2 A b+7 a^3 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^2 (11 a B+15 A b) \sin (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))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}","\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 \left(21 a^2 A b+7 a^3 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(21 a^2 A b+7 a^3 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^2 (11 a B+15 A b) \sin (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))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(2*(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]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(15*A*b + 11*a*B)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*(9*a^2*A + 26*A*b^2 + 33*a*b*B)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",10,8,33,0.2424,1,"{4025, 4074, 4047, 3769, 3771, 2639, 4045, 2641}"
415,1,277,0,1.0141731,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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^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 (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}","-\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^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 (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,"(2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*d) + (2*a^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) - (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*b^3*d) + (2*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*d) + (2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d)","A",11,9,33,0.2727,1,"{4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
416,1,210,0,0.7133202,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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}","\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,"(-2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b*d) - (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) + (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)","A",10,9,33,0.2727,1,"{4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
417,1,126,0,0.4006318,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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}","\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*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",7,7,33,0.2121,1,"{4033, 4106, 3849, 2805, 12, 3771, 2639}"
418,1,101,0,0.1982303,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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)}","\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*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)","A",5,5,33,0.1515,1,"{4038, 3771, 2641, 3849, 2805}"
419,1,149,0,0.2570619,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),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)}{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}","-\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,"(2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)","A",7,7,33,0.2121,1,"{4038, 3771, 2639, 3848, 2803, 2641, 2805}"
420,1,196,0,0.467163,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\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 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 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}","\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 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 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}",1,"(-2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(a^2*A + 3*A*b^2 - 3*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",9,8,33,0.2424,1,"{4034, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
421,1,242,0,0.7553258,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),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 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 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(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 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 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(3*a^2*A + 5*A*b^2 - 5*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(a^2 + 3*b^2)*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*d) + (2*b^3*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",10,9,33,0.2727,1,"{4034, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
422,1,406,0,1.1616634,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\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(3 a^2 A b-5 a^3 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^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(3 a^2 A b-5 a^3 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(3 a^2 A b-5 a^3 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}","\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(3 a^2 A b-5 a^3 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^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(3 a^2 A b-5 a^3 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(3 a^2 A b-5 a^3 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,"-(((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) - (a*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) + ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",11,9,33,0.2727,1,"{4029, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
423,1,315,0,0.8367065,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\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(a^2 A b-3 a^3 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}","\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(a^2 A b-3 a^3 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,"((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4029, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
424,1,257,0,0.5273129,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\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^2 A b+a^3 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}","\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^2 A b+a^3 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,"-(((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d)) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,33,0.2424,1,"{4029, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
425,1,263,0,0.5061962,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","-\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(3 a^2 A b+a^3 (-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}","-\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(3 a^2 A b+a^3 (-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,"((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((2*a^2*A - A*b^2 - a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,33,0.2424,1,"{4027, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
426,1,283,0,0.569037,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\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(4 a^2 A b-2 a^3 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{\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{b \left(5 a^2 A b-3 a^3 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}","\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(4 a^2 A b-2 a^3 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{\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{b \left(5 a^2 A b-3 a^3 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*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - ((4*a^2*A*b - 3*A*b^3 - 2*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + (b*(5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,33,0.2424,1,"{4030, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
427,1,365,0,0.8593379,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\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(16 a^2 A b^2+2 a^4 A-12 a^3 b B+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)}-\frac{\left(4 a^2 A b-2 a^3 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(7 a^2 A b-5 a^3 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{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(16 a^2 A b^2+2 a^4 A-12 a^3 b B+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)}-\frac{\left(4 a^2 A b-2 a^3 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(7 a^2 A b-5 a^3 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}",1,"-(((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) + ((2*a^4*A + 16*a^2*A*b^2 - 15*A*b^4 - 12*a^3*b*B + 9*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) - (b^2*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4030, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
428,1,583,0,1.7794141,"\int \frac{\sec ^{\frac{9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\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(3 a^2 A b-7 a^3 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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-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}","\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(3 a^2 A b-7 a^3 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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-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,"-((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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + ((15*a^4*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)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",12,10,33,0.3030,1,"{4029, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
429,1,480,0,1.3782305,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\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(a^2 A b-5 a^3 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(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(a^2 A b-5 a^3 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(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-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}","\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(a^2 A b-5 a^3 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(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(a^2 A b-5 a^3 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(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-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,"((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{4029, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
430,1,402,0,0.9140289,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\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(a^2 A b+3 a^3 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(3 a^2 A b+a^3 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(a^2 A b+3 a^3 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(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+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}","\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(a^2 A b+3 a^3 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(3 a^2 A b+a^3 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(a^2 A b+3 a^3 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(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+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,"((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4029, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
431,1,402,0,0.9140717,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(3 a^2 A b+a^3 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{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(7 a^2 A b-3 a^3 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(5 a^2 A b+a^3 (-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(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-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}","\frac{\left(3 a^2 A b+a^3 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{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(7 a^2 A b-3 a^3 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(5 a^2 A b+a^3 (-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(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-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,"-((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4029, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
432,1,402,0,0.865092,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(7 a^2 A b-3 a^3 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{(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(-5 a^2 A b^2+8 a^4 A-7 a^3 b B+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(9 a^2 A b-5 a^3 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(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+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}","-\frac{\left(7 a^2 A b-3 a^3 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{(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(-5 a^2 A b^2+8 a^4 A-7 a^3 b B+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(9 a^2 A b-5 a^3 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(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+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,"((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4*A - 5*a^2*A*b^2 + 3*A*b^4 - 7*a^3*b*B + a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4027, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
433,1,427,0,0.998036,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{b \left(11 a^2 A b-7 a^3 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{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{\left(-33 a^2 A b^3+24 a^4 A b+5 a^3 b^2 B-8 a^5 B-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{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-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{b \left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-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}","\frac{b \left(11 a^2 A b-7 a^3 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{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{\left(-33 a^2 A b^3+24 a^4 A b+5 a^3 b^2 B-8 a^5 B-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{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-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{b \left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-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,"((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((24*a^4*A*b - 33*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + (b*(35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{4030, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
434,1,521,0,1.4439798,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{b \left(13 a^2 A b-9 a^3 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{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{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-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(128 a^4 A b^2-223 a^2 A b^4+8 a^6 A+99 a^3 b^3 B-72 a^5 b B-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}-\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-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(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-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{b \left(13 a^2 A b-9 a^3 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{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{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-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(128 a^4 A b^2-223 a^2 A b^4+8 a^6 A+99 a^3 b^3 B-72 a^5 b B-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}-\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-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(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-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}",1,"-((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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^6*A + 128*a^4*A*b^2 - 223*a^2*A*b^4 + 105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b^2*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{4030, 4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
435,1,336,0,1.107808,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),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}","\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,"((4*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b + a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d) + (B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",13,13,35,0.3714,1,"{4031, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
436,1,253,0,0.7825073,"\int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{(2 a A+b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(a B+2 A b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{B \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{(2 a A+b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(a B+2 A b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{B \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,35,0.3429,1,"{4031, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
437,1,208,0,0.5433595,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],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)}{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)}}","\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*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",11,11,35,0.3143,1,"{4037, 3854, 3858, 2663, 2661, 3859, 2807, 2805, 3856, 2655, 2653}"
438,1,201,0,0.4792025,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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)}}","\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*A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4032, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
439,1,267,0,0.7479347,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","-\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)}","-\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*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4032, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
440,1,343,0,1.0349082,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),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(19 a^2 A b+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \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(19 a^2 A b+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4032, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
441,1,421,0,1.5967292,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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(6 a^2 A b+a^3 (-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}","\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(6 a^2 A b+a^3 (-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,"((42*a*A*b + 17*a^2*B + 16*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((30*a*A*b + 3*a^2*B + 16*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d) + ((6*A*b + 7*a*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (b*B*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",14,13,35,0.3714,1,"{4026, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
442,1,339,0,1.2071969,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"((8*a^2*A + 4*A*b^2 + 7*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + 5*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b + 5*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",13,13,35,0.3714,1,"{4026, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
443,1,272,0,0.8679081,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"((2*a*A*b + 2*a^2*B + b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A - b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,35,0.3429,1,"{4026, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
444,1,276,0,0.9179577,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),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)}}","\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,"(2*(a^2*A - 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)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",12,12,35,0.3429,1,"{4025, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
445,1,266,0,0.7916229,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\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)}","\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^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)]*Sqrt[Sec[c + d*x]])/(15*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
446,1,342,0,1.1285638,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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(82 a^2 A b+63 a^3 B+21 a b^2 B-6 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+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)}","\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(82 a^2 A b+63 a^3 B+21 a b^2 B-6 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+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,"(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(8*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
447,1,427,0,1.4952174,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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(88 a^2 A b+75 a^3 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(39 a^2 A b+75 a^3 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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (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)}","\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(88 a^2 A b+75 a^3 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(39 a^2 A b+75 a^3 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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (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,"(2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sec[c + d*x]^(3/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sqrt[Sec[c + d*x]])","A",11,9,35,0.2571,1,"{4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
448,1,513,0,1.9984575,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(264 a^2 A b+15 a^3 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(472 a^2 A b+133 a^3 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(264 a^2 A b+15 a^3 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(40 a^3 A b+120 a^2 b^2 B-5 a^4 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}","\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(264 a^2 A b+15 a^3 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(472 a^2 A b+133 a^3 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(264 a^2 A b+15 a^3 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(40 a^3 A b+120 a^2 b^2 B-5 a^4 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,"((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*d*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (b*(8*A*b + 11*a*B)*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",15,14,35,0.4000,1,"{4026, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
449,1,422,0,1.5940442,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(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(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(30 a^2 A b+5 a^3 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}","\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(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(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(30 a^2 A b+5 a^3 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,"((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*A*b + 33*a^2*B + 16*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (b*(2*A*b + 3*a*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",14,14,35,0.4000,1,"{4026, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
450,1,359,0,1.2490481,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\left(16 a^2 A b+8 a^3 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{\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{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}","\frac{\left(16 a^2 A b+8 a^3 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{\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{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,"((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*(4*A*b + 7*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",13,13,35,0.3714,1,"{4026, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
451,1,349,0,1.2503024,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\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{\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{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)}}","\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{\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{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,"((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((14*a*A*b + 6*a^2*B - 3*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(2*a*A - 3*b*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",13,13,35,0.3714,1,"{4025, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
452,1,342,0,1.2229406,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{2 \left(8 a^2 A b+5 a^3 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 \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 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)}}","\frac{2 \left(8 a^2 A b+5 a^3 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 \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 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,"(2*(8*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)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",13,13,35,0.3714,1,"{4025, 4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
453,1,340,0,1.1550527,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\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(145 a^2 A b+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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)}","\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(145 a^2 A b+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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,"(2*(a^2 - b^2)*(25*a^2*A + 15*A*b^2 + 56*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",10,10,35,0.2857,1,"{4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
454,1,425,0,1.5181438,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\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(163 a^2 A b+75 a^3 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(114 a^2 A b+75 a^3 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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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)}","\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(163 a^2 A b+75 a^3 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(114 a^2 A b+75 a^3 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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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,"(2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(4*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",11,10,35,0.2857,1,"{4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
455,1,519,0,1.9595665,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\frac{2 \left(1145 a^2 A b+539 a^3 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(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(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+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(285 a^2 A b^2+675 a^4 A+1254 a^3 b B-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(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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)}","\frac{2 \left(1145 a^2 A b+539 a^3 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(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(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+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(285 a^2 A b^2+675 a^4 A+1254 a^3 b B-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(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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,"(2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3465*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",12,10,35,0.2857,1,"{4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
456,1,344,0,1.1112081,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],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}","-\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,"((4*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b - 3*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",13,13,35,0.3714,1,"{4033, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
457,1,256,0,0.7310767,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(2 A b-a B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \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}}}","\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,"(B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",12,12,35,0.3429,1,"{4033, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
458,1,138,0,0.3916458,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],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)}}","\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*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4036, 3858, 2663, 2661, 3859, 2807, 2805}"
459,1,150,0,0.3103663,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),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)}}","\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*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4035, 3856, 2655, 2653, 3858, 2663, 2661}"
460,1,212,0,0.4795019,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),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)}}","\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*(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)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4034, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
461,1,280,0,0.7500226,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","-\frac{2 \left(7 a^2 A b-5 a^3 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 \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 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(7 a^2 A b-5 a^3 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 \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 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*(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)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4034, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
462,1,371,0,1.2690851,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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)}}","\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,"(B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A*b - 3*a^2*B + b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",13,13,35,0.3714,1,"{4029, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
463,1,220,0,0.6253126,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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)}}","\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,"(2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b - a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4029, 4108, 3859, 2807, 2805, 21, 3856, 2655, 2653}"
464,1,215,0,0.5718669,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),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)}}","-\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*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b - a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4027, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
465,1,235,0,0.5790977,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),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)}}","\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*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4030, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
466,1,326,0,0.8356394,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\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(5 a^2 A b-3 a^3 B+6 a b^2 B-8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \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(5 a^2 A b-3 a^3 B+6 a b^2 B-8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2*A + 8*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)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4030, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
467,1,423,0,1.2221638,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\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(9 a^2 A b-5 a^3 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(12 a^2 A b-5 a^3 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(24 a^2 A b^2+9 a^4 A-25 a^3 b B+40 a b^3 B-48 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \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(9 a^2 A b-5 a^3 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(12 a^2 A b-5 a^3 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(24 a^2 A b^2+9 a^4 A-25 a^3 b B+40 a b^3 B-48 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*(12*a^2*A*b + 48*A*b^3 - 5*a^3*B - 40*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^4*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4030, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
468,1,399,0,1.3744601,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\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 \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 (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 \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)}}","\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 \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 (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 \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,"(2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",13,13,35,0.3714,1,"{4029, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
469,1,329,0,0.8422169,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(2 a^2 A b+a^3 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)}}+\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(2 a^2 A b+a^3 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)}}+\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}}}",1,"(-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2*A + A*b^2 - 4*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4029, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
470,1,346,0,0.8215231,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \left(5 a^2 A b-2 a^3 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 (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(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","-\frac{2 \left(5 a^2 A b-2 a^3 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 (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(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(3*a^2*A - 2*A*b^2 - a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (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)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4027, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
471,1,368,0,0.9368257,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 b \left(8 a^2 A b-5 a^3 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 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 \left(9 a^2 A b-3 a^3 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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b \left(8 a^2 A b-5 a^3 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 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 \left(9 a^2 A b-3 a^3 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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4030, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
472,1,472,0,1.4085695,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-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 b \left(10 a^2 A b-7 a^3 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 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 \left(16 a^2 A b^2+a^4 A-9 a^3 b B+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(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-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 b \left(10 a^2 A b-7 a^3 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 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 \left(16 a^2 A b^2+a^4 A-9 a^3 b B+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(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(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[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*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^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",10,10,35,0.2857,1,"{4030, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
473,1,588,0,1.8786386,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \left(-71 a^2 A b^2+3 a^4 A+50 a^3 b B-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 b \left(12 a^2 A b-9 a^3 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 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 \left(-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-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(116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+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(55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(-71 a^2 A b^2+3 a^4 A+50 a^3 b B-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 b \left(12 a^2 A b-9 a^3 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 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 \left(-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-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(116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+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(55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (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)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",11,10,35,0.2857,1,"{4030, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
474,0,0,0,0.1577166,"\int (a+b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Int[(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,"(Sqrt[2]*B*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(2/3), x]","A",0,0,0,0,-1,"{}"
475,0,0,0,0.1455764,"\int \sqrt[3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[(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,"(Sqrt[2]*B*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(1/3), x]","A",0,0,0,0,-1,"{}"
476,0,0,0,0.1764413,"\int \frac{A+B \sec (c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[(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,"(Sqrt[2]*B*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-1/3), x]","A",0,0,0,0,-1,"{}"
477,0,0,0,0.1620304,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[(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,"(Sqrt[2]*B*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-2/3), x]","A",0,0,0,0,-1,"{}"
478,0,0,0,0.0929956,"\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx","Int[(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,"Defer[Int][(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x]","A",0,0,0,0,-1,"{}"
479,1,544,0,1.6303841,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\frac{\sin (c+d x) \left(6 a^2 A b^2 m (m+3)+a^4 A \left(m^2+4 m+3\right)+4 a^3 b B 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(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)+a^4 B \left(m^2+6 m+8\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) \left(a^2 A b \left(5 m^2+37 m+68\right)+2 a^3 B \left(m^2+8 m+19\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{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) (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)}","-\frac{\sin (c+d x) \left(6 a^2 A b^2 m (m+3)+a^4 A \left(m^2+4 m+3\right)+4 a^3 b B 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(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)+a^4 B \left(m^2+6 m+8\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) \left(a^2 A b \left(5 m^2+37 m+68\right)+2 a^3 B \left(m^2+8 m+19\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{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) (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,"(b*(A*b^3*(8 + 6*m + m^2) + 4*a*b^2*B*(8 + 6*m + m^2) + 2*a^3*B*(19 + 8*m + m^2) + a^2*A*b*(68 + 37*m + 5*m^2))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(3 + m)*(4 + m)) + (b^2*(b^2*B*(3 + m)^2 + 2*a*A*b*(4 + m)^2 + a^2*B*(26 + 9*m + m^2))*Sec[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)*(4 + m)) + (b*(A*b*(4 + m) + a*B*(7 + m))*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(d*(4 + m)) - ((A*b^4*m*(2 + m) + 4*a*b^3*B*m*(2 + m) + 6*a^2*A*b^2*m*(3 + m) + 4*a^3*b*B*m*(3 + m) + a^4*A*(3 + 4*m + m^2))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m)*(1 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]) + ((b^4*B*(3 + 4*m + m^2) + 4*a*A*b^3*(4 + 5*m + m^2) + 6*a^2*b^2*B*(4 + 5*m + m^2) + 4*a^3*A*b*(8 + 6*m + m^2) + a^4*B*(8 + 6*m + m^2))*Hypergeometric2F1[1/2, -m/2, (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2])","A",9,7,31,0.2258,1,"{4026, 4096, 4076, 4047, 3772, 2643, 4046}"
480,1,366,0,0.7862535,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\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(3 a^2 A b (m+2)+a^3 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 \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{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)}","-\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(3 a^2 A b (m+2)+a^3 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 \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{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,"(b*(b^2*B*(2 + m) + 3*a*A*b*(3 + m) + 2*a^2*B*(4 + m))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(3 + m)) + (b^2*(A*b*(3 + m) + a*B*(5 + m))*Sec[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(d*(3 + m)) - ((b^3*B*m*(2 + m) + 3*a*A*b^2*m*(3 + m) + 3*a^2*b*B*m*(3 + m) + a^3*A*(3 + 4*m + m^2))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(3 + m)*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((A*b^3*(1 + m) + 3*a*b^2*B*(1 + m) + 3*a^2*A*b*(2 + m) + a^3*B*(2 + m))*Hypergeometric2F1[1/2, -m/2, (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",8,6,31,0.1935,1,"{4026, 4076, 4047, 3772, 2643, 4046}"
481,1,261,0,0.4062084,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\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)}","-\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,"(b*(A*b*(2 + m) + a*B*(3 + m))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(2 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(d*(2 + m)) - ((A*b^2*m + 2*a*b*B*m + a^2*A*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((b^2*B*(1 + m) + a*(2*A*b + a*B)*(2 + m))*Hypergeometric2F1[1/2, -m/2, (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",7,5,31,0.1613,1,"{4026, 4047, 3772, 2643, 4046}"
482,1,177,0,0.2011024,"\int \sec ^m(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sec[c + d*x]^m*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","-\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)}","-\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,"(b*B*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)) - ((b*B*m + a*A*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((A*b + a*B)*Hypergeometric2F1[1/2, -m/2, (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*Sqrt[Sin[c + d*x]^2])","A",6,4,29,0.1379,1,"{3997, 3787, 3772, 2643}"
483,1,132,0,0.229494,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(6*a*(A + B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",8,7,31,0.2258,1,"{2954, 2968, 3023, 2748, 2635, 2641, 2639}"
484,1,101,0,0.2096032,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(2*a*(3*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",7,7,31,0.2258,1,"{2954, 2968, 3023, 2748, 2639, 2635, 2641}"
485,1,70,0,0.185505,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{2954, 2968, 3023, 2748, 2641, 2639}"
486,1,66,0,0.1941151,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),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)}}","\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,"(2*a*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/d + (2*a*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,31,0.1935,1,"{2954, 2968, 3021, 2748, 2641, 2639}"
487,1,95,0,0.2173694,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"(-2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,31,0.2258,1,"{2954, 2968, 3021, 2748, 2636, 2639, 2641}"
488,1,132,0,0.2321892,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"(-2*a*(5*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,31,0.2258,1,"{2954, 2968, 3021, 2748, 2636, 2641, 2639}"
489,1,194,0,0.4039347,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(4*a^2*(8*A + 9*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 6*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^2*(8*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(11*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(9*d)","A",9,8,33,0.2424,1,"{2954, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
490,1,161,0,0.3621773,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(4*a^2*(3*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(9*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",8,8,33,0.2424,1,"{2954, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
491,1,126,0,0.3474563,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",7,7,33,0.2121,1,"{2954, 2976, 2968, 3023, 2748, 2641, 2639}"
492,1,116,0,0.3367891,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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)}}","\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,"(4*a^2*A*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,33,0.2121,1,"{2954, 2975, 2968, 3023, 2748, 2641, 2639}"
493,1,120,0,0.3504513,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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)}","\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,"(-4*a^2*B*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(3*A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(3*A + 5*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,7,33,0.2121,1,"{2954, 2975, 2968, 3021, 2748, 2641, 2639}"
494,1,159,0,0.3808155,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"(-4*a^2*(5*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(5*A + 7*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 4*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,8,33,0.2424,1,"{2954, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
495,1,194,0,0.415113,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"(-4*a^2*(4*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(7*A + 9*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 6*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(4*A + 3*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",9,8,33,0.2424,1,"{2954, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
496,1,157,0,0.2646197,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(3*(7*A - 5*B)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - (5*(A - B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - (5*(A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{2954, 2977, 2748, 2635, 2641, 2639}"
497,1,124,0,0.2444923,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\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}","\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*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{2954, 2977, 2748, 2639, 2635, 2641}"
498,1,88,0,0.2202889,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\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)}","-\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*A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{2954, 2977, 2748, 2641, 2639}"
499,1,83,0,0.2224526,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","\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)}","\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,"-(((A - B)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{2954, 2978, 2748, 2641, 2639}"
500,1,113,0,0.2396352,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\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)}","\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,"((A - 3*B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - 3*B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{2954, 2978, 2748, 2636, 2639, 2641}"
501,1,152,0,0.2596199,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\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)}}","-\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*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((3*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A - 5*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{2954, 2978, 2748, 2636, 2641, 2639}"
502,1,204,0,0.4189148,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(7*(8*A - 5*B)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(3*A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(3*A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (7*(8*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A - 2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,6,33,0.1818,1,"{2954, 2977, 2748, 2635, 2641, 2639}"
503,1,171,0,0.4005949,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\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}","\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*A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (5*(2*A - B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (5*(2*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,33,0.1818,1,"{2954, 2977, 2748, 2639, 2635, 2641}"
504,1,137,0,0.3746047,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,5,33,0.1515,1,"{2954, 2977, 2748, 2641, 2639}"
505,1,121,0,0.3457243,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"-((A*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,33,0.1818,1,"{2954, 2977, 2978, 2748, 2641, 2639}"
506,1,121,0,0.355074,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"(B*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,5,33,0.1515,1,"{2954, 2978, 2748, 2641, 2639}"
507,1,164,0,0.3966397,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"((A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A - 4*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((2*A - 5*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",7,6,33,0.1818,1,"{2954, 2978, 2748, 2636, 2639, 2641}"
508,1,197,0,0.4275852,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2),x]","-\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}","-\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,"-(((4*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) - (5*(A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A - 2*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + ((4*A - 7*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((4*A - 7*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",8,6,33,0.1818,1,"{2954, 2978, 2748, 2636, 2641, 2639}"
509,1,221,0,0.5821916,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(-7*(17*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (7*(17*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{2954, 2977, 2748, 2639, 2635, 2641}"
510,1,188,0,0.547795,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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*A - 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",7,5,33,0.1515,1,"{2954, 2977, 2748, 2641, 2639}"
511,1,182,0,0.5390134,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","\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}","\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*A + B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{2954, 2977, 2978, 2748, 2641, 2639}"
512,1,178,0,0.5220842,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"-((A - B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{2954, 2977, 2978, 2748, 2641, 2639}"
513,1,180,0,0.5326983,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"((A + 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,5,33,0.1515,1,"{2954, 2978, 2748, 2641, 2639}"
514,1,221,0,0.5773529,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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*A - 49*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((9*A - 49*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((3*A - 13*B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{2954, 2978, 2748, 2636, 2639, 2641}"
515,1,220,0,0.475591,"\int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(32*a*(8*A + 9*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(8*A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(8*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(8*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]])","A",6,4,35,0.1143,1,"{2955, 4015, 3805, 3804}"
516,1,175,0,0.4033484,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(16*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(6*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,35,0.1143,1,"{2955, 4015, 3805, 3804}"
517,1,130,0,0.3316707,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}}","\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,"(4*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{2955, 4015, 3805, 3804}"
518,1,82,0,0.2622722,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}","\frac{2 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*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",3,3,35,0.08571,1,"{2955, 4013, 3804}"
519,1,96,0,0.25894,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),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}","\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[a]*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{2955, 4015, 3801, 215}"
520,1,98,0,0.2565681,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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}}","\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[a]*(2*A + B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*B*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{2955, 4016, 3801, 215}"
521,1,151,0,0.3313589,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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}}","\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[a]*(4*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*A + 3*B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2955, 4016, 3803, 3801, 215}"
522,1,196,0,0.3957947,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),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}}","\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,"(Sqrt[a]*(6*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2955, 4016, 3803, 3801, 215}"
523,1,275,0,0.7135915,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(32*a^2*(168*A + 187*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(168*A + 187*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(168*A + 187*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(168*A + 187*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(12*A + 11*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(11*d)","A",7,5,35,0.1429,1,"{2955, 4017, 4015, 3805, 3804}"
524,1,228,0,0.6863584,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(16*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(34*A + 39*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(10*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",6,5,35,0.1429,1,"{2955, 4017, 4015, 3805, 3804}"
525,1,181,0,0.6165173,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(4*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",5,5,35,0.1429,1,"{2955, 4017, 4015, 3805, 3804}"
526,1,131,0,0.3831106,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(8*a^2*(3*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",4,4,35,0.1143,1,"{2955, 4013, 3809, 3804}"
527,1,145,0,0.444454,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[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 (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^{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 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 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^{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 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^(3/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{2955, 4017, 4015, 3801, 215}"
528,1,144,0,0.433127,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(a^(3/2)*(2*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(2*A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2955, 4018, 4015, 3801, 215}"
529,1,153,0,0.4535537,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^(3/2)*(12*A + 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(4*A + 5*B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))","A",5,5,35,0.1429,1,"{2955, 4018, 4016, 3801, 215}"
530,1,200,0,0.5402091,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^(3/2)*(14*A + 11*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(6*A + 7*B)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(14*A + 11*B)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))","A",6,6,35,0.1714,1,"{2955, 4018, 4016, 3803, 3801, 215}"
531,1,247,0,0.633667,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),x]","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}","\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^{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 B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^(3/2)*(88*A + 75*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(8*A + 9*B)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))","A",7,6,35,0.1714,1,"{2955, 4018, 4016, 3803, 3801, 215}"
532,1,275,0,0.8311675,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[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 (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^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 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}","\frac{2 a^2 (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^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 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(16*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(710*A + 803*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(194*A + 209*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(14*A + 11*B)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",7,5,35,0.1429,1,"{2955, 4017, 4015, 3805, 3804}"
533,1,228,0,0.7579202,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[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 (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^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 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}","\frac{2 a^2 (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^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 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}",1,"(4*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(124*A + 135*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(4*A + 3*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)","A",6,5,35,0.1429,1,"{2955, 4017, 4015, 3805, 3804}"
534,1,178,0,0.4582485,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(64*a^3*(5*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",5,4,35,0.1143,1,"{2955, 4013, 3809, 3804}"
535,1,192,0,0.618572,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[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 (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^{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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{2 a^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^{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 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^(5/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 35*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",6,5,35,0.1429,1,"{2955, 4017, 4015, 3801, 215}"
536,1,197,0,0.6261549,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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{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{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}","\frac{a^3 (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{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{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(2*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(14*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(2*A - 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,35,0.1714,1,"{2955, 4017, 4018, 4015, 3801, 215}"
537,1,200,0,0.6270952,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt{\cos (c+d x)}}","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"(a^(5/2)*(20*A + 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(4*A - 9*B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(4*A + 7*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",6,5,35,0.1429,1,"{2955, 4018, 4015, 3801, 215}"
538,1,200,0,0.6513451,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^(5/2)*(38*A + 25*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(54*A + 49*B)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,5,35,0.1429,1,"{2955, 4018, 4016, 3801, 215}"
539,1,247,0,0.7663479,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^(5/2)*(200*A + 163*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(104*A + 95*B)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(200*A + 163*B)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))","A",7,6,35,0.1714,1,"{2955, 4018, 4016, 3803, 3801, 215}"
540,1,294,0,0.8565779,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),x]","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)}","\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^{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 B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^(5/2)*(326*A + 283*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(170*A + 157*B)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(10*A + 13*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))","A",8,6,35,0.1714,1,"{2955, 4018, 4016, 3803, 3801, 215}"
541,1,250,0,0.8432379,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",7,5,35,0.1429,1,"{2955, 4022, 4013, 3808, 206}"
542,1,207,0,0.6324014,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2955, 4022, 4013, 3808, 206}"
543,1,162,0,0.4471052,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2955, 4022, 4013, 3808, 206}"
544,1,119,0,0.2861397,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\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}","\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[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",4,4,35,0.1143,1,"{2955, 4013, 3808, 206}"
545,1,140,0,0.3429732,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\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}","\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*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)","A",6,6,35,0.1714,1,"{2955, 4023, 3808, 206, 3801, 215}"
546,1,181,0,0.5030867,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}}","-\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,"((2*A - B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",7,7,35,0.2000,1,"{2955, 4021, 4023, 3808, 206, 3801, 215}"
547,1,230,0,0.7003166,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\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}}","\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,"-((4*A - 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((4*A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,7,35,0.2000,1,"{2955, 4021, 4023, 3808, 206, 3801, 215}"
548,1,270,0,0.8663463,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((15*A - 11*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((147*A - 95*B)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2955, 4020, 4022, 4013, 3808, 206}"
549,1,223,0,0.6866668,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((11*A - 7*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A - 15*B)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,35,0.1714,1,"{2955, 4020, 4022, 4013, 3808, 206}"
550,1,176,0,0.489672,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((7*A - 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2955, 4020, 4013, 3808, 206}"
551,1,127,0,0.31737,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",4,4,35,0.1143,1,"{2955, 4012, 3808, 206}"
552,1,185,0,0.5311804,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"(2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",7,7,35,0.2000,1,"{2955, 4019, 4023, 3808, 206, 3801, 215}"
553,1,237,0,0.7443468,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"((2*A - 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) - ((A - 3*B)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,35,0.2286,1,"{2955, 4019, 4021, 4023, 3808, 206, 3801, 215}"
554,1,287,0,0.9473496,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"-((12*A - 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) + ((9*A - 13*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) - ((A - 2*B)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((6*A - 7*B)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,35,0.2286,1,"{2955, 4019, 4021, 4023, 3808, 206, 3801, 215}"
555,1,317,0,1.122913,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"-((283*A - 163*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A - 1495*B)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A - 85*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,35,0.1714,1,"{2955, 4020, 4022, 4013, 3808, 206}"
556,1,270,0,0.9099437,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"((163*A - 75*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A - 147*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((95*A - 39*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2955, 4020, 4022, 4013, 3808, 206}"
557,1,223,0,0.6995825,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"-((75*A - 19*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2955, 4020, 4013, 3808, 206}"
558,1,223,0,0.7115663,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","-\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{(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{(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}}","-\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{(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{(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,"((19*A + 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) - ((9*A - B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",6,6,35,0.1714,1,"{2955, 4019, 4020, 4013, 3808, 206}"
559,1,176,0,0.4034053,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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}}","\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,"((5*A + 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,5,35,0.1429,1,"{2955, 4012, 3810, 3808, 206}"
560,1,234,0,0.7138946,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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}}","\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,"(2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((3*A - 11*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",8,7,35,0.2000,1,"{2955, 4019, 4023, 3808, 206, 3801, 215}"
561,1,286,0,0.9578105,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\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{(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{(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}}","-\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{(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{(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,"((2*A - 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 15*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) - ((11*A - 35*B)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,35,0.2286,1,"{2955, 4019, 4021, 4023, 3808, 206, 3801, 215}"
562,1,140,0,0.2310403,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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,"(6*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a*A + 7*b*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*a*A + 7*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",8,7,31,0.2258,1,"{2954, 2968, 3023, 2748, 2635, 2641, 2639}"
563,1,108,0,0.2136193,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*a*A + 5*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",7,7,31,0.2258,1,"{2954, 2968, 3023, 2748, 2639, 2635, 2641}"
564,1,75,0,0.1930812,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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}","\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*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(a*A + 3*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{2954, 2968, 3023, 2748, 2641, 2639}"
565,1,71,0,0.1932143,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\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)}}","\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])/d + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/d + (2*b*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,31,0.1935,1,"{2954, 2968, 3021, 2748, 2641, 2639}"
566,1,103,0,0.2121934,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],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)}","\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*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a*A + b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,31,0.2258,1,"{2954, 2968, 3021, 2748, 2636, 2639, 2641}"
567,1,140,0,0.231375,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),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)}","\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,"(-2*(5*a*A + 3*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*a*A + 3*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,31,0.2258,1,"{2954, 2968, 3021, 2748, 2636, 2641, 2639}"
568,1,182,0,0.3681019,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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,"(2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",7,7,33,0.2121,1,"{2954, 2990, 3023, 2748, 2639, 2635, 2641}"
569,1,140,0,0.3322823,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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}","\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*a^2*A + 5*b*(A*b + 2*a*B))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(7*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",6,6,33,0.1818,1,"{2954, 2990, 3023, 2748, 2641, 2639}"
570,1,121,0,0.317962,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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)}}","\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*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{2954, 2988, 3023, 2748, 2641, 2639}"
571,1,126,0,0.3351612,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\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)}","\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*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2954, 2988, 3021, 2748, 2641, 2639}"
572,1,172,0,0.3748057,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],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)}","\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,"(-2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,7,33,0.2121,1,"{2954, 2988, 3021, 2748, 2636, 2639, 2641}"
573,1,214,0,0.3961289,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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)}","\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*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b^2*B*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,33,0.2121,1,"{2954, 2988, 3021, 2748, 2636, 2641, 2639}"
574,1,182,0,0.8552982,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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 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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a 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 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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"(2*(3*a^2*A + 5*A*b^2 - 5*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*(a^2 + 3*b^2)*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(3*a^4*d) + (2*b^3*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)","A",8,8,33,0.2424,1,"{2954, 2990, 3049, 3059, 2639, 3002, 2641, 2805}"
575,1,136,0,0.5861283,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\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 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 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a 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 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 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(-2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(a^2*A + 3*A*b^2 - 3*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*a^3*d) - (2*b^2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",7,7,33,0.2121,1,"{2954, 2990, 3059, 2639, 3002, 2641, 2805}"
576,1,89,0,0.2758829,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\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}","-\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,"(2*A*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(a^2*d) + (2*b*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d)","A",6,6,33,0.1818,1,"{2954, 3002, 2639, 2803, 2641, 2805}"
577,1,61,0,0.2185432,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])),x]","\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)}","\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*EllipticF[(c + d*x)/2, 2])/(a*d) - (2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d)","A",4,4,33,0.1212,1,"{2954, 3002, 2641, 2805}"
578,1,86,0,0.3931831,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\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)}}","\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*B*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) + (2*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2954, 3000, 3059, 2639, 12, 2805}"
579,1,150,0,0.8356721,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","-\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)}","-\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*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*b*d) - (2*a*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d) + (2*B*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) + (2*(A*b - a*B)*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])","A",8,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
580,1,217,0,1.1884558,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]","\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 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) \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)}","\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 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) \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,"(2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) + (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(3*b^2*d) + (2*a^2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*B*Sin[c + d*x])/(5*b*d*Cos[c + d*x]^(5/2)) + (2*(A*b - a*B)*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(3/2)) - (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sin[c + d*x])/(5*b^3*d*Sqrt[Cos[c + d*x]])","A",9,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
581,1,305,0,1.0179951,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(16 a^2 A b^2+2 a^4 A-12 a^3 b B+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)}-\frac{\left(4 a^2 A b-2 a^3 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(7 a^2 A b-5 a^3 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{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(16 a^2 A b^2+2 a^4 A-12 a^3 b B+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)}-\frac{\left(4 a^2 A b-2 a^3 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(7 a^2 A b-5 a^3 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{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)}",1,"-(((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d)) + ((2*a^4*A + 16*a^2*A*b^2 - 15*A*b^4 - 12*a^3*b*B + 9*a*b^3*B)*EllipticF[(c + d*x)/2, 2])/(3*a^4*(a^2 - b^2)*d) - (b^2*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 2989, 3049, 3059, 2639, 3002, 2641, 2805}"
582,1,223,0,0.7021948,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","-\frac{\left(4 a^2 A b-2 a^3 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{\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 \left(5 a^2 A b-3 a^3 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}+\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(4 a^2 A b-2 a^3 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{\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 \left(5 a^2 A b-3 a^3 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}+\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)}",1,"((2*a^2*A - 3*A*b^2 + a*b*B)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) - ((4*a^2*A*b - 3*A*b^3 - 2*a^3*B + a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + (b*(5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2954, 2989, 3059, 2639, 3002, 2641, 2805}"
583,1,203,0,0.6135706,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\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{\left(3 a^2 A b+a^3 (-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}-\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(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{\left(3 a^2 A b+a^3 (-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}-\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)}",1,"((A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) + ((2*a^2*A - A*b^2 - a*b*B)*EllipticF[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) - ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2954, 2999, 3059, 2639, 3002, 2641, 2805}"
584,1,197,0,0.666893,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","-\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{\left(a^2 A b+a^3 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}+\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{(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{\left(a^2 A b+a^3 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}+\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)}",1,"-(((A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) + ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2954, 3000, 3059, 2639, 3002, 2641, 2805}"
585,1,255,0,0.9441418,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2),x]","\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(a^2 A b-3 a^3 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}-\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{(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(a^2 A b-3 a^3 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}-\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)}",1,"((a*A*b - 3*a^2*B + 2*b^2*B)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) + ((A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
586,1,346,0,1.3106209,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2),x]","-\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{\left(3 a^2 A b-5 a^3 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(3 a^2 A b-5 a^3 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{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(3 a^2 A b-5 a^3 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)}}","-\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{\left(3 a^2 A b-5 a^3 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(3 a^2 A b-5 a^3 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{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(3 a^2 A b-5 a^3 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,"-(((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d)) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*b^2*(a^2 - b^2)*d) - (a*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((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)*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))","A",9,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
587,1,461,0,1.5846448,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(128 a^4 A b^2-223 a^2 A b^4+8 a^6 A+99 a^3 b^3 B-72 a^5 b B-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}-\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-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(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-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{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(13 a^2 A b-9 a^3 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(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-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(128 a^4 A b^2-223 a^2 A b^4+8 a^6 A+99 a^3 b^3 B-72 a^5 b B-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}-\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-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(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-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{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(13 a^2 A b-9 a^3 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(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-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}",1,"-((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)*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^6*A + 128*a^4*A*b^2 - 223*a^2*A*b^4 + 105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B)*EllipticF[(c + d*x)/2, 2])/(12*a^5*(a^2 - b^2)^2*d) - (b^2*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",9,9,33,0.2727,1,"{2954, 2989, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
588,1,367,0,1.1080548,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(-33 a^2 A b^3+24 a^4 A b+5 a^3 b^2 B-8 a^5 B-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{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-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{b \left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-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}+\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(11 a^2 A b-7 a^3 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(-33 a^2 A b^3+24 a^4 A b+5 a^3 b^2 B-8 a^5 B-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{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-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{b \left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-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}+\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(11 a^2 A b-7 a^3 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)}",1,"((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((24*a^4*A*b - 33*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B)*EllipticF[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + (b*(35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 2989, 3047, 3059, 2639, 3002, 2641, 2805}"
589,1,346,0,1.1018596,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\left(-5 a^2 A b^2+8 a^4 A-7 a^3 b B+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(9 a^2 A b-5 a^3 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(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+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}-\frac{\left(9 a^2 A b-5 a^3 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{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^2 A b^2+8 a^4 A-7 a^3 b B+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(9 a^2 A b-5 a^3 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(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+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}-\frac{\left(9 a^2 A b-5 a^3 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{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}",1,"((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4*A - 5*a^2*A*b^2 + 3*A*b^4 - 7*a^3*b*B + a*b^3*B)*EllipticF[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 2989, 3055, 3059, 2639, 3002, 2641, 2805}"
590,1,338,0,1.0062006,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","-\frac{\left(7 a^2 A b-3 a^3 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(5 a^2 A b+a^3 (-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(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-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}+\frac{\left(5 a^2 A b+a^3 (-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{(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(7 a^2 A b-3 a^3 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(5 a^2 A b+a^3 (-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(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-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}+\frac{\left(5 a^2 A b+a^3 (-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{(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}",1,"-((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 2999, 3055, 3059, 2639, 3002, 2641, 2805}"
591,1,342,0,1.096245,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\left(3 a^2 A b+a^3 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(a^2 A b+3 a^3 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{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+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}-\frac{a \left(a^2 A b+3 a^3 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{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(3 a^2 A b+a^3 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(a^2 A b+3 a^3 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{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+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}-\frac{a \left(a^2 A b+3 a^3 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{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}",1,"((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - (a*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
592,1,420,0,1.4778492,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\left(a^2 A b-5 a^3 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{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-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}-\frac{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 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{a \left(a^2 A b-5 a^3 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{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(a^2 A b-5 a^3 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{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 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(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-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}-\frac{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 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{a \left(a^2 A b-5 a^3 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{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}",1,"((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((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*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))","A",9,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
593,1,523,0,1.979702,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3),x]","-\frac{\left(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-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{a \left(3 a^2 A b-7 a^3 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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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{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{\left(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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)}}","-\frac{\left(15 a^3 A b+61 a^2 b^2 B-35 a^4 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(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-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{a \left(3 a^2 A b-7 a^3 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(15 a^3 A b+61 a^2 b^2 B-35 a^4 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{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{\left(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-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,"-((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)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*EllipticF[(c + d*x)/2, 2])/(12*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) + ((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*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))","A",10,8,33,0.2424,1,"{2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
594,1,343,0,1.2188842,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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(19 a^2 A b+63 a^3 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}","\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(19 a^2 A b+63 a^3 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,"(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",11,10,35,0.2857,1,"{2955, 4032, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
595,1,267,0,0.913874,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","-\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}","-\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*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,35,0.2857,1,"{2955, 4032, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
596,1,201,0,0.6203395,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\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}","\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*A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,35,0.2571,1,"{2955, 4032, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
597,1,208,0,0.6832062,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",12,12,35,0.3429,1,"{2955, 4037, 3854, 3858, 2663, 2661, 3859, 2807, 2805, 3856, 2655, 2653}"
598,1,253,0,0.9351463,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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}}}","\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,"((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,35,0.3714,1,"{2955, 4031, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
599,1,336,0,1.2620339,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"((4*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*A*b + a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4031, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
600,1,427,0,1.7077201,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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(88 a^2 A b+75 a^3 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(39 a^2 A b+75 a^3 B-18 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(33 a^2 A b^2+147 a^4 A+246 a^3 b B-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}","\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(88 a^2 A b+75 a^3 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(39 a^2 A b+75 a^3 B-18 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(33 a^2 A b^2+147 a^4 A+246 a^3 b B-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,"(2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(10*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",12,10,35,0.2857,1,"{2955, 4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
601,1,342,0,1.3042925,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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(82 a^2 A b+63 a^3 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}","\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(82 a^2 A b+63 a^3 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,"(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(8*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",11,10,35,0.2857,1,"{2955, 4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
602,1,266,0,0.970073,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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}","\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,"(2*(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)])/(15*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(6*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,35,0.2857,1,"{2955, 4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
603,1,276,0,1.0932864,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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)}}","\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,"(2*(a^2*A - 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)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",13,13,35,0.3714,1,"{2955, 4025, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
604,1,272,0,1.0228816,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\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)}}","\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,"((2*a*A*b + 2*a^2*B + b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,35,0.3714,1,"{2955, 4026, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
605,1,339,0,1.4175777,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"((8*a^2*A + 4*A*b^2 + 7*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4026, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
606,1,421,0,1.8019872,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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(6 a^2 A b+a^3 (-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)}","\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(6 a^2 A b+a^3 (-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,"((42*a*A*b + 17*a^2*B + 16*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + ((6*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Cos[c + d*x]^(3/2)) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]])","A",15,14,35,0.4000,1,"{2955, 4026, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
607,1,519,0,2.1739138,"\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(1145 a^2 A b+539 a^3 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(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+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(285 a^2 A b^2+675 a^4 A+1254 a^3 b B-110 a b^3 B+40 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-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}","\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(1145 a^2 A b+539 a^3 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(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+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(285 a^2 A b^2+675 a^4 A+1254 a^3 b B-110 a b^3 B+40 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-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,"(2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",13,11,35,0.3143,1,"{2955, 4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
608,1,425,0,1.7155521,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(163 a^2 A b+75 a^3 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(114 a^2 A b+75 a^3 B+45 a b^2 B-10 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 A b^2+147 a^4 A+435 a^3 b B+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}","\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(163 a^2 A b+75 a^3 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(114 a^2 A b+75 a^3 B+45 a b^2 B-10 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 A b^2+147 a^4 A+435 a^3 b B+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,"(2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(4*A*b + 3*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)","A",12,11,35,0.3143,1,"{2955, 4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
609,1,340,0,1.3224476,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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(145 a^2 A b+63 a^3 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}","\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(145 a^2 A b+63 a^3 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,"(2*(a^2 - b^2)*(25*a^2*A + 15*A*b^2 + 56*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(10*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)","A",11,11,35,0.3143,1,"{2955, 4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
610,1,342,0,1.393079,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 \left(8 a^2 A b+5 a^3 B+10 a b^2 B-8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\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 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)}}","\frac{2 \left(8 a^2 A b+5 a^3 B+10 a b^2 B-8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\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 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,"(2*(8*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)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(8*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",14,14,35,0.4000,1,"{2955, 4025, 4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
611,1,349,0,1.4256794,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\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{\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{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}","\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{\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{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,"((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(2*a*A - 3*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",14,14,35,0.4000,1,"{2955, 4025, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
612,1,359,0,1.4288333,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\left(16 a^2 A b+8 a^3 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{\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{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)}}","\frac{\left(16 a^2 A b+8 a^3 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{\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{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,"((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(4*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4026, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
613,1,422,0,1.795158,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\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(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(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(30 a^2 A b+5 a^3 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)}","\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(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(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(30 a^2 A b+5 a^3 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,"((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(2*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",15,15,35,0.4286,1,"{2955, 4026, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
614,1,513,0,2.2482863,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\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(264 a^2 A b+15 a^3 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(472 a^2 A b+133 a^3 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(264 a^2 A b+15 a^3 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(40 a^3 A b+120 a^2 b^2 B-5 a^4 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)}","\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(264 a^2 A b+15 a^3 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(472 a^2 A b+133 a^3 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(264 a^2 A b+15 a^3 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(40 a^3 A b+120 a^2 b^2 B-5 a^4 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,"((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(192*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(64*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(8*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(3/2)) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))","A",16,15,35,0.4286,1,"{2955, 4026, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
615,1,280,0,0.9160415,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \left(7 a^2 A b-5 a^3 B-10 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \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 (4 A b-5 a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}","-\frac{2 \left(7 a^2 A b-5 a^3 B-10 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \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 (4 A b-5 a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(-2*(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)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)","A",10,10,35,0.2857,1,"{2955, 4034, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
616,1,212,0,0.6321402,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],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}","\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^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)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",9,9,35,0.2571,1,"{2955, 4034, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
617,1,150,0,0.4348899,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{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)}}","\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*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",8,8,35,0.2286,1,"{2955, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
618,1,138,0,0.52561,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\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)}}","\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,"(2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",8,8,35,0.2286,1,"{2955, 4036, 3858, 2663, 2661, 3859, 2807, 2805}"
619,1,256,0,0.8886876,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{(2 A b-a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \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}}}","\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,"(B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",13,13,35,0.3714,1,"{2955, 4033, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
620,1,344,0,1.2811901,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","-\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)}","-\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,"((4*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) + ((4*A*b - 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4033, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
621,1,423,0,1.4044127,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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(9 a^2 A b-5 a^3 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(12 a^2 A b-5 a^3 B-40 a b^2 B+48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(24 a^2 A b^2+9 a^4 A-25 a^3 b B+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}}}","\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(9 a^2 A b-5 a^3 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(12 a^2 A b-5 a^3 B-40 a b^2 B+48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(24 a^2 A b^2+9 a^4 A-25 a^3 b B+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,"(-2*(12*a^2*A*b + 48*A*b^3 - 5*a^3*B - 40*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)","A",11,10,35,0.2857,1,"{2955, 4030, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
622,1,326,0,1.03192,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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(5 a^2 A b-3 a^3 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}}}","\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(5 a^2 A b-3 a^3 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,"(2*(a^2*A + 8*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)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)","A",10,10,35,0.2857,1,"{2955, 4030, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
623,1,235,0,0.7195835,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\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)}}","\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*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{2955, 4030, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
624,1,215,0,0.6651576,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","-\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)}}","-\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*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",9,9,35,0.2571,1,"{2955, 4027, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
625,1,220,0,0.7875577,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\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)}}","\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,"(2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,35,0.2857,1,"{2955, 4029, 4108, 3859, 2807, 2805, 21, 3856, 2655, 2653}"
626,1,371,0,1.4462422,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\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)}}","\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,"(B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4029, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
627,1,487,0,1.8587274,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]","-\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(12 a^2 A b-15 a^3 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(12 a^2 A b-15 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)}{4 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\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{(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)}}","-\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(12 a^2 A b-15 a^3 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(12 a^2 A b-15 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)}{4 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\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{(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,"((4*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a*A*b - 15*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 5*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",15,14,35,0.4000,1,"{2955, 4029, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
628,1,588,0,2.0668586,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(-71 a^2 A b^2+3 a^4 A+50 a^3 b B-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 b \left(12 a^2 A b-9 a^3 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 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 \left(-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-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(116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+80 a b^4 B-128 A b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-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}}}","\frac{2 \left(-71 a^2 A b^2+3 a^4 A+50 a^3 b B-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 b \left(12 a^2 A b-9 a^3 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 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 \left(-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-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(116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+80 a b^4 B-128 A b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-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,"(-2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^5*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d)","A",12,11,35,0.3143,1,"{2955, 4030, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
629,1,472,0,1.5401547,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-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 b \left(10 a^2 A b-7 a^3 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 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 \left(16 a^2 A b^2+a^4 A-9 a^3 b B+8 a b^3 B-16 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-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}}}","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-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 b \left(10 a^2 A b-7 a^3 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 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 \left(16 a^2 A b^2+a^4 A-9 a^3 b B+8 a b^3 B-16 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-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,"(2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",11,11,35,0.3143,1,"{2955, 4030, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
630,1,368,0,1.105226,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 b \left(8 a^2 A b-5 a^3 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 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 \left(9 a^2 A b-3 a^3 B+2 a b^2 B-8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-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}}}","\frac{2 b \left(8 a^2 A b-5 a^3 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 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 \left(9 a^2 A b-3 a^3 B+2 a b^2 B-8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-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,"(-2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,35,0.2857,1,"{2955, 4030, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
631,1,346,0,1.0076933,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","-\frac{2 \left(5 a^2 A b-2 a^3 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 (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(6 a^2 A b-3 a^3 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}}}","-\frac{2 \left(5 a^2 A b-2 a^3 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 (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(6 a^2 A b-3 a^3 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,"(2*(3*a^2*A - 2*A*b^2 - a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,35,0.2857,1,"{2955, 4027, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
632,1,329,0,1.0426169,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \left(2 a^2 A b+a^3 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)}}+\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(2 a^2 A b+a^3 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)}}+\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}}}",1,"(-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,35,0.2857,1,"{2955, 4029, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
633,1,399,0,1.5004338,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\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 \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 (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 \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)}}","\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 \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 (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 \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,"(2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",14,14,35,0.4000,1,"{2955, 4029, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
634,1,526,0,1.9902786,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 a \left(2 a^2 A b-5 a^3 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{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(6 a^3 A b+26 a^2 b^2 B-15 a^4 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(-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{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 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)}}","\frac{2 a \left(2 a^2 A b-5 a^3 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{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(6 a^3 A b+26 a^2 b^2 B-15 a^4 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(-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{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 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,"-((2*a*A*b - 5*a^2*B + 3*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])","A",15,15,35,0.4286,1,"{2955, 4029, 4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"