1,1,81,87,0.3452544,"\int \sec ^6(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^6*(A + C*Sec[c + d*x]^2),x]","\frac{A \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \left(\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{(7 A+6 C) \tan ^5(c+d x)}{35 d}+\frac{2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac{(7 A+6 C) \tan (c+d x)}{7 d}+\frac{C \tan (c+d x) \sec ^6(c+d x)}{7 d}",1,"(A*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d + (C*(Tan[c + d*x] + Tan[c + d*x]^3 + (3*Tan[c + d*x]^5)/5 + Tan[c + d*x]^7/7))/d","A",1
2,1,75,98,0.3883244,"\int \sec ^5(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2),x]","\frac{3 (6 A+5 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(2 (6 A+5 C) \sec ^2(c+d x)+3 (6 A+5 C)+8 C \sec ^4(c+d x)\right)}{48 d}","\frac{(6 A+5 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{(6 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(6 A+5 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{C \tan (c+d x) \sec ^5(c+d x)}{6 d}",1,"(3*(6*A + 5*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(3*(6*A + 5*C) + 2*(6*A + 5*C)*Sec[c + d*x]^2 + 8*C*Sec[c + d*x]^4)*Tan[c + d*x])/(48*d)","A",1
3,1,61,65,0.2168842,"\int \sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2),x]","\frac{A \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{(5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \tan (c+d x)}{5 d}+\frac{C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(A*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d + (C*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
4,1,54,70,0.1426196,"\int \sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2),x]","\frac{(4 A+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(4 A+2 C \sec ^2(c+d x)+3 C\right)}{8 d}","\frac{(4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(4*A + 3*C + 2*C*Sec[c + d*x]^2)*Tan[c + d*x])/(8*d)","A",1
5,1,36,43,0.1128992,"\int \sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2),x]","\frac{A \tan (c+d x)}{d}+\frac{C \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{(3 A+2 C) \tan (c+d x)}{3 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(A*Tan[c + d*x])/d + (C*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
6,1,48,40,0.0154827,"\int \sec (c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(A + C*Sec[c + d*x]^2),x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(A*ArcTanh[Sin[c + d*x]])/d + (C*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
7,1,15,15,0.0040161,"\int \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[A + C*Sec[c + d*x]^2,x]","A x+\frac{C \tan (c+d x)}{d}","A x+\frac{C \tan (c+d x)}{d}",1,"A*x + (C*Tan[c + d*x])/d","A",1
8,1,35,24,0.0174586,"\int \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(A + C*Sec[c + d*x]^2),x]","\frac{A \sin (c) \cos (d x)}{d}+\frac{A \cos (c) \sin (d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{A \sin (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(C*ArcTanh[Sin[c + d*x]])/d + (A*Cos[d*x]*Sin[c])/d + (A*Cos[c]*Sin[d*x])/d","A",1
9,1,33,31,0.0372321,"\int \cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2),x]","\frac{A (c+d x)}{2 d}+\frac{A \sin (2 (c+d x))}{4 d}+C x","\frac{A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (A+2 C)",1,"C*x + (A*(c + d*x))/(2*d) + (A*Sin[2*(c + d*x)])/(4*d)","A",1
10,1,50,30,0.0202274,"\int \cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2),x]","-\frac{A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d}","\frac{(A+C) \sin (c+d x)}{d}-\frac{A \sin ^3(c+d x)}{3 d}",1,"(C*Cos[d*x]*Sin[c])/d + (C*Cos[c]*Sin[d*x])/d + (A*Sin[c + d*x])/d - (A*Sin[c + d*x]^3)/(3*d)","A",1
11,1,45,61,0.0965203,"\int \cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2),x]","\frac{4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x))}{32 d}","\frac{(3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} x (3 A+4 C)",1,"(4*(3*A + 4*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + A*Sin[4*(c + d*x)])/(32*d)","A",1
12,1,71,50,0.0244244,"\int \cos ^5(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2),x]","\frac{A \sin ^5(c+d x)}{5 d}-\frac{2 A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","-\frac{(2 A+C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{A \sin ^5(c+d x)}{5 d}",1,"(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (2*A*Sin[c + d*x]^3)/(3*d) - (C*Sin[c + d*x]^3)/(3*d) + (A*Sin[c + d*x]^5)/(5*d)","A",1
13,1,68,89,0.1115824,"\int \cos ^6(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2),x]","\frac{(45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x))+60 A c+60 A d x+72 c C+72 C d x}{192 d}","\frac{(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{1}{16} x (5 A+6 C)",1,"(60*A*c + 72*c*C + 60*A*d*x + 72*C*d*x + (45*A + 48*C)*Sin[2*(c + d*x)] + (9*A + 6*C)*Sin[4*(c + d*x)] + A*Sin[6*(c + d*x)])/(192*d)","A",1
14,1,26,26,0.3523909,"\int \sec ^m(c+d x) \left(-\frac{C m}{1+m}+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(-((C*m)/(1 + m)) + C*Sec[c + d*x]^2),x]","\frac{C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)}","\frac{C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)}",1,"(C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m))","A",1
15,1,25,25,0.3384044,"\int \sec ^m(c+d x) \left(A-\frac{A (1+m) \sec ^2(c+d x)}{m}\right) \, dx","Integrate[Sec[c + d*x]^m*(A - (A*(1 + m)*Sec[c + d*x]^2)/m),x]","-\frac{A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m}","-\frac{A \sin (c+d x) \sec ^{m+1}(c+d x)}{d m}",1,"-((A*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*m))","A",1
16,1,84,110,1.3823135,"\int (b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{(b \sec (c+d x))^{7/2} \left(4 (7 A+5 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) ((7 A+5 C) \cos (2 (c+d x))+7 A+11 C)\right)}{42 b d}","\frac{2 b^2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 b (7 A+5 C) \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d}+\frac{2 C \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}",1,"((b*Sec[c + d*x])^(7/2)*(4*(7*A + 5*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(7*A + 11*C + (7*A + 5*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*b*d)","A",1
17,1,184,110,1.4359812,"\int (b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{4 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{3/2} \left((5 A+3 C) \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \left(5 A \left(1+e^{2 i (c+d x)}\right)^2+C \left(8 e^{2 i (c+d x)}+3 e^{4 i (c+d x)}+1\right)\right)\right) \left(A+C \sec ^2(c+d x)\right)}{15 d \left(1+e^{2 i (c+d x)}\right)^2 (A \cos (2 (c+d x))+A+2 C)}","-\frac{2 b^2 (5 A+3 C) 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 (5 A+3 C) \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}+\frac{2 C \tan (c+d x) (b \sec (c+d x))^{3/2}}{5 d}",1,"(((4*I)/15)*E^(I*(c + d*x))*Cos[c + d*x]^3*(-3*(5*A*(1 + E^((2*I)*(c + d*x)))^2 + C*(1 + 8*E^((2*I)*(c + d*x)) + 3*E^((4*I)*(c + d*x)))) + (5*A + 3*C)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/(d*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
18,1,58,72,0.4059412,"\int \sqrt{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{2 (b \sec (c+d x))^{3/2} \left((3 A+C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (c+d x)\right)}{3 b d}","\frac{2 (3 A+C) \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 C \tan (c+d x) \sqrt{b \sec (c+d x)}}{3 d}",1,"(2*(b*Sec[c + d*x])^(3/2)*((3*A + C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + C*Sin[c + d*x]))/(3*b*d)","A",1
19,1,126,68,0.7605533,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]","-\frac{2 i \left(2 (A-C) e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \left(A e^{2 i (c+d x)}+A-2 C e^{2 i (c+d x)}\right)\right)}{3 d \left(1+e^{2 i (c+d x)}\right) \sqrt{b \sec (c+d x)}}","\frac{2 (A-C) 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 C \tan (c+d x)}{d \sqrt{b \sec (c+d x)}}",1,"(((-2*I)/3)*(-3*(A + A*E^((2*I)*(c + d*x)) - 2*C*E^((2*I)*(c + d*x))) + 2*(A - C)*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(d*(1 + E^((2*I)*(c + d*x)))*Sqrt[b*Sec[c + d*x]])","C",1
20,1,66,75,0.3159141,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) \left(2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (2 (c+d x))\right)}{3 d (b \sec (c+d x))^{3/2}}","\frac{2 (A+3 C) \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 \tan (c+d x)}{3 d (b \sec (c+d x))^{3/2}}",1,"(Sec[c + d*x]^2*(2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[2*(c + d*x)]))/(3*d*(b*Sec[c + d*x])^(3/2))","A",1
21,1,133,77,1.31194,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2),x]","\frac{e^{-i d x} \sec ^2(c+d x) (\cos (d x)+i \sin (d x)) \left(-\frac{8 i (3 A+5 C) e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+6 A \sin (2 (c+d x))+12 i (3 A+5 C)\right)}{30 d (b \sec (c+d x))^{5/2}}","\frac{2 (3 A+5 C) 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 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])*((12*I)*(3*A + 5*C) - ((8*I)*(3*A + 5*C)*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 6*A*Sin[2*(c + d*x)]))/(30*d*E^(I*d*x)*(b*Sec[c + d*x])^(5/2))","C",1
22,1,79,112,0.7470545,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2),x]","\frac{2 \sin (c+d x) (3 A \cos (2 (c+d x))+13 A+14 C)+\frac{4 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)}}}{42 b^3 d \sqrt{b \sec (c+d x)}}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 b^4 d}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}",1,"((4*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 2*(13*A + 14*C + 3*A*Cos[2*(c + d*x)])*Sin[c + d*x])/(42*b^3*d*Sqrt[b*Sec[c + d*x]])","A",1
23,1,143,112,1.5564198,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2),x]","\frac{e^{-i d x} (\cos (d x)+i \sin (d x)) \left(-\frac{32 i (7 A+9 C) e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+(76 A+72 C) \sin (2 (c+d x))+10 A \sin (4 (c+d x))+336 i A+432 i C\right)}{360 b^4 d \sqrt{b \sec (c+d x)}}","\frac{2 (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 b^4 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 (7 A+9 C) \sin (c+d x)}{45 b^3 d (b \sec (c+d x))^{3/2}}+\frac{2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}",1,"((Cos[d*x] + I*Sin[d*x])*((336*I)*A + (432*I)*C - ((32*I)*(7*A + 9*C)*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + (76*A + 72*C)*Sin[2*(c + d*x)] + 10*A*Sin[4*(c + d*x)]))/(360*b^4*d*E^(I*d*x)*Sqrt[b*Sec[c + d*x]])","C",1
24,1,21,21,0.1748657,"\int \frac{3+3 \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(3 + 3*Sec[c + d*x]^2)/Sqrt[Sec[c + d*x]],x]","\frac{6 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}","\frac{6 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(6*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",1
25,1,21,21,0.3285243,"\int \sec ^m(e+f x) \left(m-(1+m) \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^m*(m - (1 + m)*Sec[e + f*x]^2),x]","-\frac{\sin (e+f x) \sec ^{m+1}(e+f x)}{f}","-\frac{\sin (e+f x) \sec ^{m+1}(e+f x)}{f}",1,"-((Sec[e + f*x]^(1 + m)*Sin[e + f*x])/f)","A",1
26,1,19,19,0.0686462,"\int \sec ^5(e+f x) \left(5-6 \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^5*(5 - 6*Sec[e + f*x]^2),x]","-\frac{\tan (e+f x) \sec ^5(e+f x)}{f}","-\frac{\tan (e+f x) \sec ^5(e+f x)}{f}",1,"-((Sec[e + f*x]^5*Tan[e + f*x])/f)","A",1
27,1,19,19,0.0366776,"\int \sec ^4(e+f x) \left(4-5 \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^4*(4 - 5*Sec[e + f*x]^2),x]","-\frac{\tan (e+f x) \sec ^4(e+f x)}{f}","-\frac{\tan (e+f x) \sec ^4(e+f x)}{f}",1,"-((Sec[e + f*x]^4*Tan[e + f*x])/f)","A",1
28,1,19,19,0.0383108,"\int \sec ^3(e+f x) \left(3-4 \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^3*(3 - 4*Sec[e + f*x]^2),x]","-\frac{\tan (e+f x) \sec ^3(e+f x)}{f}","-\frac{\tan (e+f x) \sec ^3(e+f x)}{f}",1,"-((Sec[e + f*x]^3*Tan[e + f*x])/f)","A",1
29,1,19,19,0.0629195,"\int \sec ^2(e+f x) \left(2-3 \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^2*(2 - 3*Sec[e + f*x]^2),x]","-\frac{\tan (e+f x) \sec ^2(e+f x)}{f}","-\frac{\tan (e+f x) \sec ^2(e+f x)}{f}",1,"-((Sec[e + f*x]^2*Tan[e + f*x])/f)","A",1
30,1,17,17,0.0170419,"\int \sec (e+f x) \left(1-2 \sec ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]*(1 - 2*Sec[e + f*x]^2),x]","-\frac{\tan (e+f x) \sec (e+f x)}{f}","-\frac{\tan (e+f x) \sec (e+f x)}{f}",1,"-((Sec[e + f*x]*Tan[e + f*x])/f)","A",1
31,1,11,11,0.0051119,"\int -\sec ^2(e+f x) \, dx","Integrate[-Sec[e + f*x]^2,x]","-\frac{\tan (e+f x)}{f}","-\frac{\tan (e+f x)}{f}",1,"-(Tan[e + f*x]/f)","A",1
32,1,23,11,0.0083233,"\int -\cos (e+f x) \, dx","Integrate[-Cos[e + f*x],x]","-\frac{\sin (e) \cos (f x)}{f}-\frac{\cos (e) \sin (f x)}{f}","-\frac{\sin (e+f x)}{f}",1,"-((Cos[f*x]*Sin[e])/f) - (Cos[e]*Sin[f*x])/f","B",1
33,1,33,17,0.014697,"\int \cos ^2(e+f x) \left(-2+\sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^2*(-2 + Sec[e + f*x]^2),x]","-\frac{\sin (2 e) \cos (2 f x)}{2 f}-\frac{\cos (2 e) \sin (2 f x)}{2 f}","-\frac{\sin (e+f x) \cos (e+f x)}{f}",1,"-1/2*(Cos[2*f*x]*Sin[2*e])/f - (Cos[2*e]*Sin[2*f*x])/(2*f)","A",1
34,1,46,19,0.0409921,"\int \cos ^3(e+f x) \left(-3+2 \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^3*(-3 + 2*Sec[e + f*x]^2),x]","\frac{\sin ^3(e+f x)}{f}-\frac{3 \sin (e+f x)}{f}+\frac{2 \sin (e) \cos (f x)}{f}+\frac{2 \cos (e) \sin (f x)}{f}","-\frac{\sin (e+f x) \cos ^2(e+f x)}{f}",1,"(2*Cos[f*x]*Sin[e])/f + (2*Cos[e]*Sin[f*x])/f - (3*Sin[e + f*x])/f + Sin[e + f*x]^3/f","B",1
35,1,31,19,0.0306983,"\int \cos ^4(e+f x) \left(-4+3 \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^4*(-4 + 3*Sec[e + f*x]^2),x]","-\frac{\sin (2 (e+f x))}{4 f}-\frac{\sin (4 (e+f x))}{8 f}","-\frac{\sin (e+f x) \cos ^3(e+f x)}{f}",1,"-1/4*Sin[2*(e + f*x)]/f - Sin[4*(e + f*x)]/(8*f)","A",1
36,1,38,19,0.0325765,"\int \cos ^5(e+f x) \left(-5+4 \sec ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^5*(-5 + 4*Sec[e + f*x]^2),x]","-\frac{\sin ^5(e+f x)}{f}+\frac{2 \sin ^3(e+f x)}{f}-\frac{\sin (e+f x)}{f}","-\frac{\sin (e+f x) \cos ^4(e+f x)}{f}",1,"-(Sin[e + f*x]/f) + (2*Sin[e + f*x]^3)/f - Sin[e + f*x]^5/f","A",1
37,1,76,85,0.2757938,"\int \sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 C \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}+\frac{3 C \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 C \tan (c+d x) \sec (c+d x)}{8 d}",1,"(C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*C*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (B*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
38,1,60,63,0.1642719,"\int \sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (C*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
39,1,47,47,0.0185205,"\int \sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
40,1,24,24,0.0125834,"\int \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[B*Sec[c + d*x] + C*Sec[c + d*x]^2,x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/d + (C*Tan[c + d*x])/d","A",1
41,1,16,16,0.0040773,"\int \cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}",1,"B*x + (C*ArcTanh[Sin[c + d*x]])/d","A",1
42,1,26,15,0.0087399,"\int \cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \sin (c) \cos (d x)}{d}+\frac{B \cos (c) \sin (d x)}{d}+C x","\frac{B \sin (c+d x)}{d}+C x",1,"C*x + (B*Cos[d*x]*Sin[c])/d + (B*Cos[c]*Sin[d*x])/d","A",1
43,1,35,38,0.0686957,"\int \cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B (2 (c+d x)+\sin (2 (c+d x)))+4 C \sin (c+d x)}{4 d}","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x)}{d}",1,"(4*C*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d)","A",1
44,1,57,54,0.0690491,"\int \cos ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C (c+d x)}{2 d}+\frac{C \sin (2 (c+d x))}{4 d}","-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}",1,"(C*(c + d*x))/(2*d) + (B*Sin[c + d*x])/d - (B*Sin[c + d*x]^3)/(3*d) + (C*Sin[2*(c + d*x)])/(4*d)","A",1
45,1,73,76,0.133003,"\int \cos ^5(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 B (c+d x)}{8 d}+\frac{B \sin (2 (c+d x))}{4 d}+\frac{B \sin (4 (c+d x))}{32 d}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","\frac{B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 B x}{8}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}",1,"(3*B*(c + d*x))/(8*d) + (C*Sin[c + d*x])/d - (C*Sin[c + d*x]^3)/(3*d) + (B*Sin[2*(c + d*x)])/(4*d) + (B*Sin[4*(c + d*x)])/(32*d)","A",1
46,1,89,92,0.118013,"\int \cos ^6(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{3 C (c+d x)}{8 d}+\frac{C \sin (2 (c+d x))}{4 d}+\frac{C \sin (4 (c+d x))}{32 d}","\frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 C x}{8}",1,"(3*C*(c + d*x))/(8*d) + (B*Sin[c + d*x])/d - (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d) + (C*Sin[2*(c + d*x)])/(4*d) + (C*Sin[4*(c + d*x)])/(32*d)","A",1
47,1,102,169,0.5055256,"\int (b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(b \sec (c+d x))^{5/2} \left(10 B \sin (2 (c+d x))+20 B \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 C \sin (c+d x)+9 C \sin (3 (c+d x))-36 C \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 b d}","-\frac{6 b^2 C 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))^{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 C \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 b d}+\frac{6 b C \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}",1,"((b*Sec[c + d*x])^(5/2)*(-36*C*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*B*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*C*Sin[c + d*x] + 10*B*Sin[2*(c + d*x)] + 9*C*Sin[3*(c + d*x)]))/(30*b*d)","A",1
48,1,90,135,0.3053155,"\int \sqrt{b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(b \sec (c+d x))^{3/2} \left(2 \sin (c+d x) (3 B \cos (c+d x)+C)-6 B \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 C \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 b 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 C \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 b d}+\frac{2 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 d}",1,"((b*Sec[c + d*x])^(3/2)*(-6*B*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*C*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(C + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(3*b*d)","A",1
49,1,73,109,0.2048594,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]","\frac{2 \left(B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{C \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d \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 C \sin (c+d x) \sqrt{b \sec (c+d x)}}{b d}-\frac{2 C 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*(-(C*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (C*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])","A",1
50,1,57,85,0.0908272,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}","\frac{2 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{b^2 d}+\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] + C*EllipticF[(c + d*x)/2, 2]))/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])","A",1
51,1,81,116,0.1843353,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{B \sin (2 (c+d x))}{\sqrt{\cos (c+d x)}}+6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 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)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}",1,"(6*C*EllipticE[(c + d*x)/2, 2] + 2*B*EllipticF[(c + d*x)/2, 2] + (B*Sin[2*(c + d*x)])/Sqrt[Cos[c + d*x]])/(3*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])","A",1
52,1,91,147,0.5799664,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (3 B \cos (c+d x)+5 C)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^4 d}","\frac{2 C \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^4 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 C \sin (c+d x)}{3 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]*(9*B*EllipticE[(c + d*x)/2, 2] + 5*C*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*C + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*b^4*d)","A",1
53,1,104,176,0.6953717,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2),x]","\frac{\sqrt{b \sec (c+d x)} \left(\sin (2 (c+d x)) (15 B \cos (2 (c+d x))+65 B+42 C \cos (c+d x))+100 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 C \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 b^5 d}","\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 b^5 d}+\frac{10 B \sin (c+d x)}{21 b^4 d \sqrt{b \sec (c+d x)}}+\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b^3 d (b \sec (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{7 b^2 d (b \sec (c+d x))^{5/2}}",1,"(Sqrt[b*Sec[c + d*x]]*(252*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*B + 42*C*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*b^5*d)","A",1
54,1,80,122,0.6527874,"\int \sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(8 \left(5 (A+2 C) \tan ^2(c+d x)+15 (A+C)+3 C \tan ^4(c+d x)\right)+30 B \sec ^3(c+d x)+45 B \sec (c+d x)\right)+45 B \tanh ^{-1}(\sin (c+d x))}{120 d}","\frac{(5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \tan (c+d x)}{5 d}+\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(45*B*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(45*B*Sec[c + d*x] + 30*B*Sec[c + d*x]^3 + 8*(15*(A + C) + 5*(A + 2*C)*Tan[c + d*x]^2 + 3*C*Tan[c + d*x]^4)))/(120*d)","A",1
55,1,71,97,0.2610135,"\int \sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(3 (4 A+3 C) \sec (c+d x)+8 B \left(\tan ^2(c+d x)+3\right)+6 C \sec ^3(c+d x)\right)+3 (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{24 d}","\frac{(4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(4*A + 3*C)*Sec[c + d*x] + 6*C*Sec[c + d*x]^3 + 8*B*(3 + Tan[c + d*x]^2)))/(24*d)","A",1
56,1,51,78,0.2354193,"\int \sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(6 (A+C)+3 B \sec (c+d x)+2 C \tan ^2(c+d x)\right)+3 B \tanh ^{-1}(\sin (c+d x))}{6 d}","\frac{(3 A+2 C) \tan (c+d x)}{3 d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*B*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + C) + 3*B*Sec[c + d*x] + 2*C*Tan[c + d*x]^2))/(6*d)","A",1
57,1,59,51,0.0194691,"\int \sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x)}{d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(A*ArcTanh[Sin[c + d*x]])/d + (C*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
58,1,27,27,0.0072674,"\int \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[A + B*Sec[c + d*x] + C*Sec[c + d*x]^2,x]","A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d}","A x+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \tan (c+d x)}{d}",1,"A*x + (B*ArcTanh[Sin[c + d*x]])/d + (C*Tan[c + d*x])/d","A",1
59,1,38,27,0.0289874,"\int \cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{A \sin (c) \cos (d x)}{d}+\frac{A \cos (c) \sin (d x)}{d}+B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{A \sin (c+d x)}{d}+B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}",1,"B*x + (C*ArcTanh[Sin[c + d*x]])/d + (A*Cos[d*x]*Sin[c])/d + (A*Cos[c]*Sin[d*x])/d","A",1
60,1,55,42,0.0611993,"\int \cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{A (c+d x)}{2 d}+\frac{A \sin (2 (c+d x))}{4 d}+\frac{B \sin (c) \cos (d x)}{d}+\frac{B \cos (c) \sin (d x)}{d}+C x","\frac{A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (A+2 C)+\frac{B \sin (c+d x)}{d}",1,"C*x + (A*(c + d*x))/(2*d) + (B*Cos[d*x]*Sin[c])/d + (B*Cos[c]*Sin[d*x])/d + (A*Sin[2*(c + d*x)])/(4*d)","A",1
61,1,53,56,0.1429133,"\int \cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (3 A+4 C) \sin (c+d x)+A \sin (3 (c+d x))+3 B \sin (2 (c+d x))+6 B c+6 B d x}{12 d}","\frac{(A+C) \sin (c+d x)}{d}-\frac{A \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}",1,"(6*B*c + 6*B*d*x + 3*(3*A + 4*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)])/(12*d)","A",1
62,1,70,88,0.2002679,"\int \cos ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{24 (A+C) \sin (2 (c+d x))+3 A \sin (4 (c+d x))+36 A c+36 A d x-32 B \sin ^3(c+d x)+96 B \sin (c+d x)+48 c C+48 C d x}{96 d}","\frac{(3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} x (3 A+4 C)-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}",1,"(36*A*c + 48*c*C + 36*A*d*x + 48*C*d*x + 96*B*Sin[c + d*x] - 32*B*Sin[c + d*x]^3 + 24*(A + C)*Sin[2*(c + d*x)] + 3*A*Sin[4*(c + d*x)])/(96*d)","A",1
63,1,87,98,0.2211531,"\int \cos ^5(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{60 (5 A+6 C) \sin (c+d x)+50 A \sin (3 (c+d x))+6 A \sin (5 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+40 C \sin (3 (c+d x))}{480 d}","-\frac{(2 A+C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{A \sin ^5(c+d x)}{5 d}+\frac{B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 B x}{8}",1,"(180*B*c + 180*B*d*x + 60*(5*A + 6*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 50*A*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)])/(480*d)","A",1
64,1,102,132,0.3481177,"\int \cos ^6(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{5 ((45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x))+60 A c+60 A d x+72 c C+72 C d x)+192 B \sin ^5(c+d x)-640 B \sin ^3(c+d x)+960 B \sin (c+d x)}{960 d}","\frac{(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{1}{16} x (5 A+6 C)+\frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}",1,"(960*B*Sin[c + d*x] - 640*B*Sin[c + d*x]^3 + 192*B*Sin[c + d*x]^5 + 5*(60*A*c + 72*c*C + 60*A*d*x + 72*C*d*x + (45*A + 48*C)*Sin[2*(c + d*x)] + (9*A + 6*C)*Sin[4*(c + d*x)] + A*Sin[6*(c + d*x)]))/(960*d)","A",1
65,1,640,178,6.5841405,"\int (b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{3 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{2 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{5 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{4 B \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{3 d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{\cos ^3(c+d x) (b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (5 A+3 C) \csc (c) \cos (d x)}{5 d}+\frac{4 \sec (c) \sec (c+d x) (5 B \sin (d x)+3 C \sin (c))}{15 d}+\frac{4 B \tan (c)}{3 d}+\frac{4 C \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}\right)}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C}","-\frac{2 b^2 (5 A+3 C) 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 (5 A+3 C) \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 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 C \tan (c+d x) (b \sec (c+d x))^{3/2}}{5 d}",1,"(4*B*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (Cos[c + d*x]^3*(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(5*A + 3*C)*Cos[d*x]*Csc[c])/(5*d) + (4*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]*(3*C*Sin[c] + 5*B*Sin[d*x]))/(15*d) + (4*B*Tan[c])/(3*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])","C",1
66,1,249,136,2.0145668,"\int \sqrt{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 \cos (c+d x) (3 B \csc (c) \cos (d x) \cos (c+d x)+C \sin (c+d x))}{d}-\frac{i e^{-3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \left(\left(-1+e^{2 i c}\right) (3 A+C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 B \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+3 B \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d}\right)}{3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 (3 A+C) \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 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 C \tan (c+d x) \sqrt{b \sec (c+d x)}}{3 d}",1,"(Sqrt[b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-I)*(1 + E^((2*I)*(c + d*x)))^(5/2)*(3*B*Sqrt[1 + E^((2*I)*(c + d*x))] + 3*B*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + (3*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*c))) + (4*Cos[c + d*x]*(3*B*Cos[d*x]*Cos[c + d*x]*Csc[c] + C*Sin[c + d*x]))/d))/(3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",1
67,1,156,110,1.5100598,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]","\frac{2 e^{-i d x} (\sin (d x)-i \cos (d x)) \sqrt{b \sec (c+d x)} \left((A-C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 A \cos (c+d x)+3 i B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 i C \sin (c+d x)+3 C \cos (c+d x)\right)}{3 b d}","\frac{2 (A-C) 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 C \tan (c+d x)}{d \sqrt{b \sec (c+d x)}}",1,"(2*Sqrt[b*Sec[c + d*x]]*((-I)*Cos[d*x] + Sin[d*x])*(-3*A*Cos[c + d*x] + 3*C*Cos[c + d*x] + (3*I)*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A - C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (3*I)*C*Sin[c + d*x]))/(3*b*d*E^(I*d*x))","C",1
68,1,179,117,1.4873866,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2),x]","-\frac{i e^{-i (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \left(A \left(-1+e^{2 i (c+d x)}\right)+6 B e^{i (c+d x)}\right)+4 (A+3 C) e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)-12 B e^{i (c+d x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right)}{3 b d \sqrt{1+e^{2 i (c+d x)}} \sqrt{b \sec (c+d x)}}","\frac{2 (A+3 C) \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 \tan (c+d x)}{3 d (b \sec (c+d x))^{3/2}}+\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,"((-1/3*I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(6*B*E^(I*(c + d*x)) + A*(-1 + E^((2*I)*(c + d*x)))) - 12*B*E^(I*(c + d*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 4*(A + 3*C)*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(b*d*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[b*Sec[c + d*x]])","C",1
69,1,169,150,2.2548044,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2),x]","\frac{e^{-i d x} (\cos (d x)+i \sin (d x)) \sqrt{b \sec (c+d x)} \left(\cos (c+d x) (3 A \sin (2 (c+d x))+6 i (3 A+5 C)+10 B \sin (c+d x))-2 i (3 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^3 d}","\frac{2 (3 A+5 C) 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 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}",1,"(Sqrt[b*Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(3*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((6*I)*(3*A + 5*C) + 10*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)])))/(15*b^3*d*E^(I*d*x))","C",1
70,1,313,185,4.4677298,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \tan (c+d x) (15 A \cos (2 (c+d x))+65 A+42 B \cos (c+d x)+70 C)-252 B \cot (c) \sec (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 i \sqrt{2} \left(-5 \left(-1+e^{2 i c}\right) (5 A+7 C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+63 B \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+63 B \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{105 (b \sec (c+d x))^{7/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 b^4 d}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}",1,"(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((4*I)*Sqrt[2]*(63*B*Sqrt[1 + E^((2*I)*(c + d*x))] + 63*B*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*(5*A + 7*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-252*B*Cot[c]*Sec[c + d*x] + 2*(65*A + 70*C + 42*B*Cos[c + d*x] + 15*A*Cos[2*(c + d*x)])*Tan[c + d*x])/(d*Sec[c + d*x]^(3/2))))/(105*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(7/2))","C",1