1,1,131,0,0.1757704,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","-\frac{a (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}","-\frac{a (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(4*A + 3*C)*x)/8 + (a*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{3034, 3023, 2748, 2635, 8, 2633}"
2,1,108,0,0.1012983,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(4*A + 3*C)*x)/8 + (a*(3*A + 2*C)*Sin[c + d*x])/(3*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",3,3,29,0.1034,1,"{3034, 3023, 2734}"
3,1,81,0,0.0635726,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (3 A+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}","\frac{a (3 A+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}",1,"(a*(2*A + C)*x)/2 + (a*(3*A + C)*Sin[c + d*x])/(3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)","A",2,2,23,0.08696,1,"{3024, 2734}"
4,1,58,0,0.1087179,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a x (2 A+C)+\frac{a C \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a x (2 A+C)+\frac{a C \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*A + C)*x)/2 + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,29,0.1379,1,"{3034, 3023, 2735, 3770}"
5,1,42,0,0.1016079,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+a C x","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+a C x",1,"a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d","A",4,4,31,0.1290,1,"{3032, 3023, 2735, 3770}"
6,1,58,0,0.1248254,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x",1,"a*C*x + (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,31,0.1290,1,"{3032, 3021, 2735, 3770}"
7,1,86,0,0.1670517,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{3032, 3021, 2748, 3767, 8, 3770}"
8,1,117,0,0.1897398,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{3032, 3021, 2748, 3768, 3770, 3767, 8}"
9,1,194,0,0.472005,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","-\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+11 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}","-\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+11 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^2*(14*A + 11*C)*x)/16 + (2*a^2*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a^2*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(10*A + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (2*a^2*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)","A",9,8,33,0.2424,1,"{3046, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
10,1,163,0,0.2902713,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac{a^2 (4 A+3 C) \sin (c+d x) \cos (c+d x)}{12 d}+\frac{1}{4} a^2 x (4 A+3 C)+\frac{(10 A+3 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{30 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{10 a d}","\frac{a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac{a^2 (4 A+3 C) \sin (c+d x) \cos (c+d x)}{12 d}+\frac{1}{4} a^2 x (4 A+3 C)+\frac{(10 A+3 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{30 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{10 a d}",1,"(a^2*(4*A + 3*C)*x)/4 + (a^2*(4*A + 3*C)*Sin[c + d*x])/(3*d) + (a^2*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d) + ((10*A + 3*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(30*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(10*a*d)","A",5,5,31,0.1613,1,"{3046, 2968, 3023, 2751, 2644}"
11,1,123,0,0.1393837,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}","\frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}",1,"(a^2*(12*A + 7*C)*x)/8 + (a^2*(12*A + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)","A",3,3,25,0.1200,1,"{3024, 2751, 2644}"
12,1,96,0,0.2980158,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"a^2*(2*A + C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Sin[c + d*x])/d + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{3046, 2976, 2968, 3023, 2735, 3770}"
13,1,112,0,0.3906401,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+3 C)+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}","-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+3 C)+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^2*(2*A + 3*C)*x)/2 + (2*a^2*A*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 3*C)*Sin[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d","A",6,6,33,0.1818,1,"{3044, 2976, 2968, 3023, 2735, 3770}"
14,1,112,0,0.3587187,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+2 a^2 C x+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}","-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+2 a^2 C x+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"2*a^2*C*x + (a^2*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) + (A*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,33,0.1818,1,"{3044, 2975, 2968, 3023, 2735, 3770}"
15,1,110,0,0.3538211,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (A+C) \tan (c+d x)}{d}+\frac{a^2 (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{a^2 (A+C) \tan (c+d x)}{d}+\frac{a^2 (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"a^2*C*x + (a^2*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Tan[c + d*x])/d + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{3044, 2975, 2968, 3021, 2735, 3770}"
16,1,147,0,0.4491688,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{2 a^2 (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}","\frac{2 a^2 (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"(a^2*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a^2*(11*A + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,8,33,0.2424,1,"{3044, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
17,1,178,0,0.4735998,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 (9 A+10 C) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{a^2 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 (9 A+10 C) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^2*(18*A + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*(9*A + 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(10*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,9,33,0.2727,1,"{3044, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
18,1,237,0,0.6067876,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","-\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(A+C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+21 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{14 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}","-\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(A+C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+21 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{14 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*(26*A + 21*C)*x)/16 + (a^3*(133*A + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(154*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(14*a*d) + ((A + C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^3*(133*A + 108*C)*Sin[c + d*x]^3)/(105*d)","A",10,8,33,0.2424,1,"{3046, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
19,1,188,0,0.3337391,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","-\frac{a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+23 C)+\frac{(30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}","-\frac{a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+23 C)+\frac{(30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}",1,"(a^3*(30*A + 23*C)*x)/16 + (a^3*(30*A + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d) - (a^3*(30*A + 23*C)*Sin[c + d*x]^3)/(120*d)","A",11,9,31,0.2903,1,"{3046, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
20,1,148,0,0.1984915,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","-\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+13 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}","-\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+13 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}",1,"(a^3*(20*A + 13*C)*x)/8 + (a^3*(20*A + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(20*A + 13*C)*Sin[c + d*x]^3)/(60*d)","A",9,7,25,0.2800,1,"{3024, 2751, 2645, 2637, 2635, 8, 2633}"
21,1,147,0,0.4388512,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac{(4 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^3 x (28 A+15 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac{(4 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^3 x (28 A+15 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(28*A + 15*C)*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 3*C)*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(4*a*d) + ((4*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(8*d)","A",7,6,31,0.1935,1,"{3046, 2976, 2968, 3023, 2735, 3770}"
22,1,145,0,0.4523359,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{(6 A-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+5 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}","-\frac{(6 A-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+5 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^3*(6*A + 5*C)*x)/2 + (3*a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d) - ((6*A - 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d","A",7,6,33,0.1818,1,"{3044, 2976, 2968, 3023, 2735, 3770}"
23,1,160,0,0.480221,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{a^3 (7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{3 A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (2 A+7 C)+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}","-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{a^3 (7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{3 A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (2 A+7 C)+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^3*(2*A + 7*C)*x)/2 + (a^3*(7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - ((4*A - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (3*A*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,33,0.2121,1,"{3044, 2975, 2976, 2968, 3023, 2735, 3770}"
24,1,156,0,0.4997176,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+3 a^3 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}","\frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+3 a^3 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"3*a^3*C*x + (a^3*(5*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + ((5*A + 3*C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/(3*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,6,33,0.1818,1,"{3044, 2975, 2968, 3023, 2735, 3770}"
25,1,169,0,0.4963962,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{5 a^3 (3 A+4 C) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(5 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+a^3 C x+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^3 (3 A+4 C) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(5 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+a^3 C x+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"a^3*C*x + (a^3*(15*A + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*C)*Tan[c + d*x])/(8*d) + ((5*A + 4*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,6,33,0.1818,1,"{3044, 2975, 2968, 3021, 2735, 3770}"
26,1,194,0,0.5722798,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^3 (38 A+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{3 A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}","\frac{a^3 (38 A+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{3 A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(a^3*(13*A + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 55*C)*Tan[c + d*x])/(15*d) + (a^3*(109*A + 140*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((11*A + 10*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (3*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,8,33,0.2424,1,"{3044, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
27,1,225,0,0.6176588,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}","\frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"(a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 30*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",10,9,33,0.2727,1,"{3044, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
28,1,279,0,0.7929159,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","-\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+323 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}","-\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+323 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}",1,"(a^4*(392*A + 323*C)*x)/128 + (4*a^4*(63*A + 52*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((56*A + 61*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(8*A + 7*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (4*a^4*(63*A + 52*C)*Sin[c + d*x]^3)/(105*d)","A",11,8,33,0.2424,1,"{3046, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
29,1,219,0,0.4120924,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","-\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac{1}{4} a^4 x (14 A+11 C)+\frac{(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d}","-\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac{1}{4} a^4 x (14 A+11 C)+\frac{(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d}",1,"(a^4*(14*A + 11*C)*x)/4 + (16*a^4*(14*A + 11*C)*Sin[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(21*a*d) - (8*a^4*(14*A + 11*C)*Sin[c + d*x]^3)/(105*d)","A",14,9,31,0.2903,1,"{3046, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
30,1,179,0,0.2323559,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","-\frac{2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}","-\frac{2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}",1,"(7*a^4*(10*A + 7*C)*x)/16 + (4*a^4*(10*A + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 7*C)*Sin[c + d*x]^3)/(15*d)","A",12,7,25,0.2800,1,"{3024, 2751, 2645, 2637, 2635, 8, 2633}"
31,1,177,0,0.5388119,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{(5 A+7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{(8 A+7 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^4 x (12 A+7 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{(5 A+7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{(8 A+7 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^4 x (12 A+7 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*(12*A + 7*C)*x)/2 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(10*A + 7*C)*Sin[c + d*x])/(2*d) + (a*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d)","A",8,6,31,0.1935,1,"{3046, 2976, 2968, 3023, 2735, 3770}"
32,1,181,0,0.602695,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{(12 A-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+35 C)-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}","\frac{5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{(12 A-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+35 C)-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}",1,"(a^4*(52*A + 35*C)*x)/8 + (4*a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 7*C)*Sin[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - ((12*A - 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - ((12*A - 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d","A",8,6,33,0.1818,1,"{3044, 2976, 2968, 3023, 2735, 3770}"
33,1,186,0,0.6076243,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}-\frac{(9 A-4 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+2 a^4 x (2 A+3 C)+\frac{2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}","-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}-\frac{(9 A-4 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+2 a^4 x (2 A+3 C)+\frac{2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}",1,"2*a^4*(2*A + 3*C)*x + (a^4*(13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - 2*C)*Sin[c + d*x])/(2*d) - ((15*A - 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((9*A - 4*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",8,7,33,0.2121,1,"{3044, 2975, 2976, 2968, 3023, 2735, 3770}"
34,1,198,0,0.6861716,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{5 a^4 (2 A-C) \sin (c+d x)}{2 d}+\frac{2 a^4 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(22 A+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{(8 A+3 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}+\frac{1}{2} a^4 x (2 A+13 C)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}+\frac{2 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{3 d}","-\frac{5 a^4 (2 A-C) \sin (c+d x)}{2 d}+\frac{2 a^4 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(22 A+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{(8 A+3 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}+\frac{1}{2} a^4 x (2 A+13 C)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}+\frac{2 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(a^4*(2*A + 13*C)*x)/2 + (2*a^4*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/d - (5*a^4*(2*A - C)*Sin[c + d*x])/(2*d) - ((22*A + 3*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + ((8*A + 3*C)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{3044, 2975, 2976, 2968, 3023, 2735, 3770}"
35,1,200,0,0.6689548,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+\frac{(7 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+4 a^4 C x+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}","-\frac{5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+\frac{(7 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+4 a^4 C x+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}",1,"4*a^4*C*x + (a^4*(35*A + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 4*C)*Sin[c + d*x])/(8*d) + ((35*A + 36*C)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(12*d) + ((7*A + 4*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,6,33,0.1818,1,"{3044, 2975, 2968, 3023, 2735, 3770}"
36,1,207,0,0.6652372,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^4 (7 A+10 C) \tan (c+d x)}{2 d}+\frac{a^4 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(7 A+5 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{(7 A+8 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+a^4 C x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{a^4 (7 A+10 C) \tan (c+d x)}{2 d}+\frac{a^4 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(7 A+5 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{(7 A+8 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+a^4 C x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"a^4*C*x + (a^4*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*(7*A + 10*C)*Tan[c + d*x])/(2*d) + ((7*A + 8*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + ((7*A + 5*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,6,33,0.1818,1,"{3044, 2975, 2968, 3021, 2735, 3770}"
37,1,232,0,0.7614683,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{4 a^4 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(37 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{(43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}+\frac{2 a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}","\frac{4 a^4 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(37 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{(43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}+\frac{2 a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}",1,"(7*a^4*(7*A + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(18*A + 25*C)*Tan[c + d*x])/(15*d) + (a^4*(417*A + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((43*A + 50*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((37*A + 30*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",10,8,33,0.2424,1,"{3044, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
38,1,263,0,0.8024215,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{210 d}+\frac{a^4 (11 A+14 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{(8 A+7 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 d}+\frac{(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{210 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{21 d}","\frac{a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{210 d}+\frac{a^4 (11 A+14 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{(8 A+7 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 d}+\frac{(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{210 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{21 d}",1,"(a^4*(11*A + 14*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^4*(454*A + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(11*A + 14*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^4*(247*A + 308*C)*Sec[c + d*x]^2*Tan[c + d*x])/(210*d) + ((109*A + 126*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(210*d) + ((8*A + 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(21*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)","A",11,9,33,0.2727,1,"{3044, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
39,1,156,0,0.1913814,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A+5 C)}{8 a}","\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A+5 C)}{8 a}",1,"(3*(4*A + 5*C)*x)/(8*a) - ((3*A + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((4*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A + 4*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,5,33,0.1515,1,"{3042, 2748, 2633, 2635, 8}"
40,1,124,0,0.173477,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A+3 C)}{2 a}","-\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A+3 C)}{2 a}",1,"-((2*A + 3*C)*x)/(2*a) + ((3*A + 4*C)*Sin[c + d*x])/(a*d) - ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - ((3*A + 4*C)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,33,0.1515,1,"{3042, 2748, 2635, 8, 2633}"
41,1,98,0,0.0935063,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(A+2 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A+3 C)}{2 a}","-\frac{(A+2 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A+3 C)}{2 a}",1,"((2*A + 3*C)*x)/(2*a) - ((A + 2*C)*Sin[c + d*x])/(a*d) + ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",2,2,31,0.06452,1,"{3042, 2734}"
42,1,48,0,0.0966013,"\int \frac{A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{(A+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{C \sin (c+d x)}{a d}-\frac{C x}{a}","\frac{(A+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{C \sin (c+d x)}{a d}-\frac{C x}{a}",1,"-((C*x)/a) + (C*Sin[c + d*x])/(a*d) + ((A + C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))","A",3,3,25,0.1200,1,"{3024, 2735, 2648}"
43,1,48,0,0.1036868,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","-\frac{(A+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}","-\frac{(A+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}",1,"(C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",3,3,31,0.09677,1,"{3042, 2735, 3770}"
44,1,61,0,0.1393447,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{(2 A+C) \tan (c+d x)}{a d}-\frac{(A+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}","\frac{(2 A+C) \tan (c+d x)}{a d}-\frac{(A+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}",1,"-((A*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A + C)*Tan[c + d*x])/(a*d) - ((A + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{3042, 2748, 3767, 8, 3770}"
45,1,105,0,0.1758342,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","-\frac{(2 A+C) \tan (c+d x)}{a d}+\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}","-\frac{(2 A+C) \tan (c+d x)}{a d}+\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A + C)*Tan[c + d*x])/(a*d) + ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3042, 2748, 3768, 3770, 3767, 8}"
46,1,133,0,0.1832442,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}","\frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"-((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((4*A + 3*C)*Tan[c + d*x])/(a*d) - ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A + 3*C)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,33,0.1515,1,"{3042, 2748, 3767, 3768, 3770}"
47,1,191,0,0.3377999,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac{2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac{(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{x (28 A+55 C)}{8 a^2}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac{2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac{(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{x (28 A+55 C)}{8 a^2}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((28*A + 55*C)*x)/(8*a^2) - (8*(A + 2*C)*Sin[c + d*x])/(a^2*d) + ((28*A + 55*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + ((28*A + 55*C)*Cos[c + d*x]^3*Sin[c + d*x])/(12*a^2*d) - (2*(A + 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (8*(A + 2*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",8,6,33,0.1818,1,"{3042, 2977, 2748, 2633, 2635, 8}"
48,1,163,0,0.3271013,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac{2 (2 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(2 A+5 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x (2 A+5 C)}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac{2 (2 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(2 A+5 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x (2 A+5 C)}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((2*A + 5*C)*x)/a^2) + ((5*A + 12*C)*Sin[c + d*x])/(a^2*d) - ((2*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (2*(2*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A + 12*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,6,33,0.1818,1,"{3042, 2977, 2748, 2635, 8, 2633}"
49,1,141,0,0.2625123,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{4 (A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{2 (A+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A+7 C)}{2 a^2}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{4 (A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{2 (A+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A+7 C)}{2 a^2}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*A + 7*C)*x)/(2*a^2) - (4*(A + 4*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*(A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,33,0.09091,1,"{3042, 2977, 2734}"
50,1,90,0,0.2410353,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}+\frac{2 C \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 C x}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}+\frac{2 C \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 C x}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*C*x)/a^2 + ((A + 4*C)*Sin[c + d*x])/(3*a^2*d) + (2*C*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,31,0.1935,1,"{3042, 2968, 3023, 12, 2735, 2648}"
51,1,66,0,0.125207,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^2 + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,25,0.1200,1,"{3020, 2735, 2648}"
52,1,77,0,0.2269675,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (2 A-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{2 (2 A-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^2*d) - (2*(2*A - C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",4,4,31,0.1290,1,"{3042, 2978, 12, 3770}"
53,1,91,0,0.2963045,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((10*A + C)*Tan[c + d*x])/(3*a^2*d) - (2*A*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 8, 3770}"
54,1,146,0,0.3158348,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{4 (4 A+C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{2 (4 A+C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{4 (4 A+C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{2 (4 A+C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(4*A + C)*Tan[c + d*x])/(3*a^2*d) + ((7*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (2*(4*A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,33,0.2121,1,"{3042, 2978, 2748, 3768, 3770, 3767, 8}"
55,1,172,0,0.3345332,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{(12 A+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A+5 C) \tan (c+d x)}{a^2 d}-\frac{(5 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(5 A+2 C) \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{2 (5 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(12 A+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A+5 C) \tan (c+d x)}{a^2 d}-\frac{(5 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(5 A+2 C) \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{2 (5 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((5*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((12*A + 5*C)*Tan[c + d*x])/(a^2*d) - ((5*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (2*(5*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((12*A + 5*C)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 3768, 3770}"
56,1,216,0,0.4848952,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A+23 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A+23 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((6*A + 23*C)*x)/(2*a^3) + (4*(9*A + 34*C)*Sin[c + d*x])/(5*a^3*d) - ((6*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) - (4*(9*A + 34*C)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,6,33,0.1818,1,"{3042, 2977, 2748, 2635, 8, 2633}"
57,1,189,0,0.4600404,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (11 A+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A+13 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{2 (11 A+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A+13 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((2*A + 13*C)*x)/(2*a^3) - (2*(11*A + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,3,33,0.09091,1,"{3042, 2977, 2734}"
58,1,136,0,0.4625825,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(2 A+27 C) \sin (c+d x)}{15 a^3 d}+\frac{3 C \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(2 A+27 C) \sin (c+d x)}{15 a^3 d}+\frac{3 C \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(-3*C*x)/a^3 + ((2*A + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*C*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3042, 2977, 2968, 3023, 12, 2735, 2648}"
59,1,114,0,0.255591,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(6 A-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(6 A-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^3 - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((6*A - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3042, 2968, 3019, 2735, 2648}"
60,1,98,0,0.120647,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{(2 A+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(2 A+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,25,0.1200,1,"{3020, 2750, 2648}"
61,1,115,0,0.3145716,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{(22 A-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{(22 A-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,4,31,0.1290,1,"{3042, 2978, 12, 3770}"
62,1,129,0,0.4427148,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 A \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 A \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(-3*A*ArcTanh[Sin[c + d*x]])/(a^3*d) + (2*(36*A + C)*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (3*A*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 8, 3770}"
63,1,192,0,0.5130089,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (76 A+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{2 (76 A+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,33,0.2121,1,"{3042, 2978, 2748, 3768, 3770, 3767, 8}"
64,1,225,0,0.5544835,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","\frac{4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-((23*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) + (4*(34*A + 9*C)*Tan[c + d*x])/(5*a^3*d) - ((23*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(34*A + 9*C)*Tan[c + d*x]^3)/(15*a^3*d)","A",8,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 3768, 3770}"
65,1,223,0,0.6120624,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","-\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{16 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A+21 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}","-\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{16 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A+21 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"((2*A + 21*C)*x)/(2*a^4) - (32*(5*A + 54*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (16*(5*A + 54*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",5,3,33,0.09091,1,"{3042, 2977, 2734}"
66,1,174,0,0.5771242,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(-4*C*x)/a^4 + (2*(3*A + 122*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*C*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,7,33,0.2121,1,"{3042, 2977, 2968, 3023, 12, 2735, 2648}"
67,1,152,0,0.4448765,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{(16 A-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(16 A-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(C*x)/a^4 - ((8*A - 55*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((16*A - 215*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(2*A - 5*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,6,33,0.1818,1,"{3042, 2977, 2968, 3019, 2735, 2648}"
68,1,138,0,0.3492955,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{4 (2 A+9 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 (3 A-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{4 (2 A+9 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 (3 A-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"((23*A - 54*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*(2*A + 9*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(3*A - 4*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",5,5,31,0.1613,1,"{3042, 2968, 3019, 2750, 2648}"
69,1,136,0,0.1709269,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 13*C)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((6*A + 13*C)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",4,4,25,0.1600,1,"{3020, 2750, 2650, 2648}"
70,1,145,0,0.4655264,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (20 A-C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (5 A-2 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","-\frac{8 (20 A-C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (5 A-2 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 8*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (8*(20*A - C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(5*A - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,4,31,0.1290,1,"{3042, 2978, 12, 3770}"
71,1,161,0,0.6163741,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 A \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 A \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-4*A*ArcTanh[Sin[c + d*x]])/(a^4*d) + (2*(332*A + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 3*C)*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*A*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(6*A - C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 8, 3770}"
72,1,224,0,0.6648693,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{32 (54 A+5 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{16 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}","-\frac{32 (54 A+5 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{16 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"((21*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(54*A + 5*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (16*(54*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",9,7,33,0.2121,1,"{3042, 2978, 2748, 3768, 3770, 3767, 8}"
73,1,257,0,0.7064486,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^4,x]","\frac{4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac{2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (11 A+2 C) \tan (c+d x) \sec (c+d x)}{a^4 d}-\frac{4 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{2 (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac{2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (11 A+2 C) \tan (c+d x) \sec (c+d x)}{a^4 d}-\frac{4 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{2 (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-2*(11*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^4*d) + (4*(454*A + 83*C)*Tan[c + d*x])/(35*a^4*d) - (2*(11*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^4*d) - ((178*A + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(11*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(8*A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (4*(454*A + 83*C)*Tan[c + d*x]^3)/(105*a^4*d)","A",9,6,33,0.1818,1,"{3042, 2978, 2748, 3767, 3768, 3770}"
74,1,223,0,0.5059402,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*(99*A + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)","A",6,6,35,0.1714,1,"{3046, 2981, 2770, 2759, 2751, 2646}"
75,1,180,0,0.4198352,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 (21 A+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (21 A+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*(21*A + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)","A",5,5,35,0.1429,1,"{3046, 2981, 2759, 2751, 2646}"
76,1,137,0,0.3051652,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 (35 A+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}","\frac{2 (35 A+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}",1,"(2*a*(35*A + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)","A",5,5,33,0.1515,1,"{3046, 2968, 3023, 2751, 2646}"
77,1,95,0,0.1279871,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}","\frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}",1,"(2*a*(15*A + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",3,3,27,0.1111,1,"{3024, 2751, 2646}"
78,1,96,0,0.2590459,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,33,0.1212,1,"{3046, 2981, 2773, 206}"
79,1,94,0,0.296169,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d","A",4,4,35,0.1143,1,"{3044, 2981, 2773, 206}"
80,1,110,0,0.3114774,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\sqrt{a} (3 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{\sqrt{a} (3 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a]*(3*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,35,0.1143,1,"{3044, 2980, 2773, 206}"
81,1,153,0,0.3880362,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (5 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{a (5 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(5*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{3044, 2980, 2772, 2773, 206}"
82,1,196,0,0.4710428,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a (35 A+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}","\frac{a (35 A+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(35*A + 48*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(35*A + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{3044, 2980, 2772, 2773, 206}"
83,1,225,0,0.638437,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac{4 a (143 A+112 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1155 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}","\frac{2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac{4 a (143 A+112 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1155 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}",1,"(2*a^2*(143*A + 112*C)*Sin[c + d*x])/(165*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(33*A + 28*C)*Cos[c + d*x]^3*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1155*d) + (2*a*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",6,6,35,0.1714,1,"{3046, 2976, 2981, 2759, 2751, 2646}"
84,1,174,0,0.3641942,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}","\frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}",1,"(8*a^2*(63*A + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)","A",6,6,33,0.1818,1,"{3046, 2968, 3023, 2751, 2647, 2646}"
85,1,132,0,0.1745629,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}","\frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}",1,"(8*a^2*(35*A + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",4,4,27,0.1481,1,"{3024, 2751, 2647, 2646}"
86,1,133,0,0.4097977,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(5*A + 4*C)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,33,0.1515,1,"{3046, 2976, 2981, 2773, 206}"
87,1,136,0,0.4409842,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a^2 (3 A-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}","-\frac{a^2 (3 A-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}",1,"(3*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(3*A - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",5,5,35,0.1429,1,"{3044, 2976, 2981, 2773, 206}"
88,1,147,0,0.4666662,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{3 a A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}","-\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{3 a A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a^(3/2)*(7*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(5*A - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,35,0.1429,1,"{3044, 2975, 2981, 2773, 206}"
89,1,155,0,0.5168465,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (19 A+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}","\frac{a^2 (19 A+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(a^(3/2)*(11*A + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(19*A + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{3044, 2975, 2980, 2773, 206}"
90,1,200,0,0.607056,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (75 A+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (13 A+16 C) \tan (c+d x) \sec (c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}","\frac{a^2 (75 A+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (13 A+16 C) \tan (c+d x) \sec (c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}",1,"(a^(3/2)*(75*A + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 16*C)*Sec[c + d*x]*Tan[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,6,35,0.1714,1,"{3044, 2975, 2980, 2772, 2773, 206}"
91,1,245,0,0.6843895,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (133 A+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (67 A+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}","\frac{a^2 (133 A+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (67 A+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}",1,"(a^(3/2)*(133*A + 176*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(133*A + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,6,35,0.1714,1,"{3044, 2975, 2980, 2772, 2773, 206}"
92,1,273,0,0.8614826,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 a^3 (2717 A+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}","\frac{2 a^3 (2717 A+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(2*a^3*(10439*A + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (10*a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)","A",7,6,35,0.1714,1,"{3046, 2976, 2981, 2759, 2751, 2646}"
93,1,211,0,0.4102162,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{16 a^2 (33 A+25 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}","\frac{16 a^2 (33 A+25 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}",1,"(64*a^3*(33*A + 25*C)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d) + (2*(99*A + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (10*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)","A",7,6,33,0.1818,1,"{3046, 2968, 3023, 2751, 2647, 2646}"
94,1,169,0,0.2081978,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{16 a^2 (21 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}","\frac{16 a^2 (21 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}",1,"(64*a^3*(21*A + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)","A",5,4,27,0.1481,1,"{3024, 2751, 2647, 2646}"
95,1,170,0,0.5669542,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(49*A + 32*C)*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(7*A + 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",6,5,33,0.1515,1,"{3046, 2976, 2981, 2773, 206}"
96,1,173,0,0.5893017,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^3 (15 A+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{5 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}","\frac{a^3 (15 A+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{5 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}",1,"(5*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*(15*A + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d","A",6,5,35,0.1429,1,"{3044, 2976, 2981, 2773, 206}"
97,1,184,0,0.6331111,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{a^{5/2} (19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{5 a A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}","-\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{a^{5/2} (19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{5 a A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}",1,"(a^(5/2)*(19*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(27*A - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,35,0.1714,1,"{3044, 2975, 2976, 2981, 2773, 206}"
98,1,192,0,0.6589648,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{5 a^{5/2} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{5 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}","-\frac{a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{5 a^{5/2} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{5 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}",1,"(5*a^(5/2)*(5*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(49*A - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,5,35,0.1429,1,"{3044, 2975, 2981, 2773, 206}"
99,1,200,0,0.7168697,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^3 (299 A+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (17 A+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{5 a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}","\frac{a^3 (299 A+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (17 A+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{5 a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^(5/2)*(163*A + 304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(299*A + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{3044, 2975, 2980, 2773, 206}"
100,1,245,0,0.7988674,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (79 A+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^3 (787 A+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}","\frac{a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (79 A+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^3 (787 A+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}",1,"(a^(5/2)*(283*A + 400*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,6,35,0.1714,1,"{3044, 2975, 2980, 2772, 2773, 206}"
101,1,290,0,0.9065084,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^3 (1015 A+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^2 (23 A+24 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^3 (109 A+136 C) \tan (c+d x) \sec ^2(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}","\frac{a^3 (1015 A+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^2 (23 A+24 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^3 (109 A+136 C) \tan (c+d x) \sec ^2(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}",1,"(a^(5/2)*(1015*A + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1015*A + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(109*A + 136*C)*Sec[c + d*x]^2*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(23*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(96*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",8,6,35,0.1714,1,"{3044, 2975, 2980, 2772, 2773, 206}"
102,1,236,0,0.8168765,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (21 A+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (21 A+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)","A",8,7,35,0.2000,1,"{3046, 2983, 2968, 3023, 2751, 2649, 206}"
103,1,193,0,0.5599014,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (35 A+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (35 A+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)","A",7,7,35,0.2000,1,"{3046, 2983, 2968, 3023, 2751, 2649, 206}"
104,1,152,0,0.331004,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (15 A+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}","\frac{2 (15 A+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}",1,"-((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)","A",6,6,33,0.1818,1,"{3046, 2968, 3023, 2751, 2649, 206}"
105,1,109,0,0.1407641,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",4,4,27,0.1481,1,"{3024, 2751, 2649, 206}"
106,1,115,0,0.2916716,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{3046, 2985, 2649, 206, 2773}"
107,1,113,0,0.3154816,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,35,0.1429,1,"{3044, 2985, 2649, 206, 2773}"
108,1,159,0,0.489791,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{(7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((7*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,35,0.1714,1,"{3044, 2984, 2985, 2649, 206, 2773}"
109,1,200,0,0.6492361,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(7 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{(7 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"-((9*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((7*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,35,0.1714,1,"{3044, 2984, 2985, 2649, 206, 2773}"
110,1,243,0,0.8431981,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(21 A+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}","-\frac{(21 A+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"((107*A + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((21*A + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,35,0.1714,1,"{3044, 2984, 2985, 2649, 206, 2773}"
111,1,259,0,0.7917157,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(245 A+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}","\frac{(245 A+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)","A",8,7,35,0.2000,1,"{3042, 2983, 2968, 3023, 2751, 2649, 206}"
112,1,214,0,0.5945379,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(5 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{(7 A+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(15 A+31 C) \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(5 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{(7 A+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(15 A+31 C) \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}",1,"-((7*A + 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((15*A + 31*C)*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)","A",7,7,35,0.2000,1,"{3042, 2983, 2968, 3023, 2751, 2649, 206}"
113,1,169,0,0.3414964,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}","\frac{(3 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"((3*A + 11*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)","A",6,6,33,0.1818,1,"{3042, 2968, 3023, 2751, 2649, 206}"
114,1,114,0,0.1438859,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}","\frac{(A-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((A - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,27,0.1481,1,"{3020, 2751, 2649, 206}"
115,1,125,0,0.3196388,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(5 A-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(5 A-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,33,0.1515,1,"{3042, 2985, 2649, 206, 2773}"
116,1,158,0,0.4914873,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(9 A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(9 A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(-3*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((9*A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,35,0.1714,1,"{3042, 2984, 2985, 2649, 206, 2773}"
117,1,217,0,0.7092541,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A + 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((7*A + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,35,0.1714,1,"{3042, 2984, 2985, 2649, 206, 2773}"
118,1,266,0,0.8811595,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(47 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(47 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}",1,"-((47*A + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*a^(3/2)*d) + ((17*A + 9*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (3*(7*A + 4*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,35,0.1714,1,"{3042, 2984, 2985, 2649, 206, 2773}"
119,1,259,0,0.8039557,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(45 A+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(195 A+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(465 A+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(45 A+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(195 A+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(465 A+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"-((75*A + 283*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)","A",8,8,35,0.2286,1,"{3042, 2977, 2983, 2968, 3023, 2751, 2649, 206}"
120,1,212,0,0.6059573,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{5 (3 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{5 (3 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A + 163*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (5*(3*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)","A",7,7,35,0.2000,1,"{3042, 2977, 2968, 3023, 2751, 2649, 206}"
121,1,165,0,0.3673332,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(A+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{5 (A-15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(A+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{5 (A-15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(5*(A - 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((3*A - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 9*C)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,33,0.1818,1,"{3042, 2968, 3019, 2751, 2649, 206}"
122,1,124,0,0.1598183,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(3 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((3*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,27,0.1481,1,"{3020, 2750, 2649, 206}"
123,1,162,0,0.4808247,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(43 A-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{(43 A-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,33,0.1818,1,"{3042, 2978, 2985, 2649, 206, 2773}"
124,1,199,0,0.6981745,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(35 A+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(115 A+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(15 A-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(35 A+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(115 A+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(15 A-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-5*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((115*A + 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,35,0.2000,1,"{3042, 2978, 2984, 2985, 2649, 206, 2773}"
125,1,262,0,0.9133858,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(63 A+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(31 A+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{(63 A+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(31 A+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((39*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A + 43*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((63*A + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,35,0.2000,1,"{3042, 2978, 2984, 2985, 2649, 206, 2773}"
126,1,196,0,0.2363909,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{10 a (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{10 a (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*a*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)","A",8,6,33,0.1818,1,"{3034, 3023, 2748, 2635, 2639, 2641}"
127,1,165,0,0.2142122,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",7,6,33,0.1818,1,"{3034, 3023, 2748, 2635, 2641, 2639}"
128,1,134,0,0.1858849,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,6,33,0.1818,1,"{3034, 3023, 2748, 2639, 2635, 2641}"
129,1,101,0,0.1638708,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,33,0.1515,1,"{3034, 3023, 2748, 2641, 2639}"
130,1,95,0,0.1680623,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,33,0.1515,1,"{3032, 3023, 2748, 2641, 2639}"
131,1,95,0,0.1715743,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,33,0.1515,1,"{3032, 3021, 2748, 2641, 2639}"
132,1,132,0,0.1985984,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{3032, 3021, 2748, 2636, 2639, 2641}"
133,1,165,0,0.2176987,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,33,0.1818,1,"{3032, 3021, 2748, 2636, 2641, 2639}"
134,1,230,0,0.4781172,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (33 A+25 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (33 A+25 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}","\frac{8 a^2 (33 A+25 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (33 A+25 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"(4*a^2*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)","A",9,8,35,0.2286,1,"{3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
135,1,197,0,0.4334294,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{4 a^2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}","\frac{4 a^2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(16*a^2*(3*A + 2*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*C*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)","A",8,8,35,0.2286,1,"{3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
136,1,164,0,0.4194662,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{8 a^2 (7 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{8 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}","\frac{8 a^2 (7 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{8 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"(4*a^2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*C*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",7,7,35,0.2000,1,"{3046, 2976, 2968, 3023, 2748, 2641, 2639}"
137,1,160,0,0.4140868,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{4 a^2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{16 a^2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}","\frac{4 a^2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{16 a^2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"(16*a^2*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(15*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",7,7,35,0.2000,1,"{3044, 2976, 2968, 3023, 2748, 2641, 2639}"
138,1,156,0,0.4245248,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{8 a^2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{8 a^2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",7,7,35,0.2000,1,"{3044, 2975, 2968, 3023, 2748, 2641, 2639}"
139,1,156,0,0.4356763,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{4 a^2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{16 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{16 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-16*a^2*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(17*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",7,7,35,0.2000,1,"{3044, 2975, 2968, 3021, 2748, 2641, 2639}"
140,1,197,0,0.4744695,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{8 a^2 (3 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{8 a^2 (3 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))","A",8,8,35,0.2286,1,"{3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
141,1,230,0,0.5068186,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{4 a^2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{4 a^2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-16*a^2*(2*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(19*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (16*a^2*(2*A + 3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))","A",9,8,35,0.2286,1,"{3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
142,1,279,0,0.6562853,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{4 a^3 (121 A+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}","\frac{4 a^3 (121 A+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}",1,"(4*a^3*(221*A + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 95*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(143*A + 118*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (12*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(143*A + 145*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)","A",10,8,35,0.2286,1,"{3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
143,1,246,0,0.6039755,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{4 a^3 (143 A+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^3 (44 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d}+\frac{2 (33 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}","\frac{4 a^3 (143 A+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^3 (44 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d}+\frac{2 (33 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (8*a^3*(44*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*C*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(33*A + 35*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d)","A",9,8,35,0.2286,1,"{3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
144,1,213,0,0.5796785,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{4 a^3 (21 A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (21 A+16 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}","\frac{4 a^3 (21 A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (21 A+16 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"(4*a^3*(27*A + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(21*A + 16*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*C*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 73*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)","A",8,7,35,0.2000,1,"{3046, 2976, 2968, 3023, 2748, 2641, 2639}"
145,1,217,0,0.5874264,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{4 a^3 (35 A+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}","\frac{4 a^3 (35 A+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"(4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^3*(35*A - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(35*A - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",8,7,35,0.2000,1,"{3044, 2976, 2968, 3023, 2748, 2641, 2639}"
146,1,211,0,0.5897304,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{4 a^3 (5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{8 a^3 (10 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{4 a^3 (5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{8 a^3 (10 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^3*(5*A - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (8*a^3*(10*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(35*A - 3*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",8,8,35,0.2286,1,"{3044, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
147,1,213,0,0.5851758,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{4 a^3 (3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (11 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^3 (3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (11 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a^3*(9*A - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(21*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)) + (2*(11*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,35,0.2000,1,"{3044, 2975, 2968, 3023, 2748, 2641, 2639}"
148,1,213,0,0.612531,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{4 a^3 (13 A+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (53 A+70 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{12 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{4 a^3 (13 A+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (53 A+70 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{12 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(53*A + 70*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (12*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",8,7,35,0.2000,1,"{3044, 2975, 2968, 3021, 2748, 2641, 2639}"
149,1,246,0,0.6363892,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{4 a^3 (11 A+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{4 a^3 (11 A+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-4*a^3*(17*A + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(73*A + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))","A",9,8,35,0.2286,1,"{3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
150,1,279,0,0.6703959,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{4 a^3 (105 A+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (35 A+33 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{4 a^3 (105 A+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (35 A+33 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(-4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^3*(35*A + 44*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(5*A + 7*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Cos[c + d*x]^(9/2)) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2))","A",10,8,35,0.2286,1,"{3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
151,1,192,0,0.2163438,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{5 (7 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}","\frac{5 (7 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(-3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,5,35,0.1429,1,"{3042, 2748, 2635, 2639, 2641}"
152,1,159,0,0.1878225,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","-\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,5,35,0.1429,1,"{3042, 2748, 2635, 2641, 2639}"
153,1,122,0,0.1723802,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"-(((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,35,0.1429,1,"{3042, 2748, 2639, 2635, 2641}"
154,1,83,0,0.1541957,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,35,0.1143,1,"{3042, 2748, 2641, 2639}"
155,1,113,0,0.1729507,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}","-\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"-(((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",5,5,35,0.1429,1,"{3042, 2748, 2636, 2639, 2641}"
156,1,150,0,0.1934214,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",6,5,35,0.1429,1,"{3042, 2748, 2636, 2641, 2639}"
157,1,192,0,0.2118472,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]","-\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}","-\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))","A",7,5,35,0.1429,1,"{3042, 2748, 2636, 2639, 2641}"
158,1,196,0,0.3506197,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{5 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{4 (5 A+14 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{4 (5 A+14 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(4*(5*A + 14*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*(5*A + 14*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,35,0.1714,1,"{3042, 2977, 2748, 2635, 2641, 2639}"
159,1,161,0,0.3330585,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 (A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((A + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (2*(A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,35,0.1714,1,"{3042, 2977, 2748, 2639, 2635, 2641}"
160,1,126,0,0.2872613,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{(A-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{4 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{4 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(4*C*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,35,0.1429,1,"{3042, 2977, 2748, 2641, 2639}"
161,1,125,0,0.2992147,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,35,0.1429,1,"{3042, 2978, 2748, 2641, 2639}"
162,1,155,0,0.323075,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{(5 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(5 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"(-4*A*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (4*A*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",6,6,35,0.1714,1,"{3042, 2978, 2748, 2636, 2639, 2641}"
163,1,189,0,0.3523236,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{2 (5 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{2 (5 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((7*A + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(5*A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(5*A + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",7,6,35,0.1714,1,"{3042, 2978, 2748, 2636, 2641, 2639}"
164,1,250,0,0.5413729,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{(13 A+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (A+6 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{(13 A+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (A+6 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(7*(7*A + 33*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A + 63*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(A + 6*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,35,0.1714,1,"{3042, 2977, 2748, 2635, 2641, 2639}"
165,1,209,0,0.494935,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","\frac{(A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"-((9*A + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 11*C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{3042, 2977, 2748, 2639, 2635, 2641}"
166,1,178,0,0.4802643,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(A-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (A-4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(A-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (A-4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((A - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,35,0.1429,1,"{3042, 2977, 2748, 2641, 2639}"
167,1,180,0,0.4691606,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}","\frac{(A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"((A - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(2*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{3042, 2977, 2978, 2748, 2641, 2639}"
168,1,184,0,0.4822343,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{(3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (3 A-2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (3 A-2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((9*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(3*A - 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,35,0.1429,1,"{3042, 2978, 2748, 2641, 2639}"
169,1,219,0,0.523739,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{(13 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (4 A-C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{(13 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (4 A-C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"-((49*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{3042, 2978, 2748, 2636, 2639, 2641}"
170,1,242,0,0.523538,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{(11 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(11 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((119*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A + C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A + C)*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",8,6,35,0.1714,1,"{3042, 2978, 2748, 2636, 2641, 2639}"
171,1,214,0,0.4747405,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{a (48 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}","\frac{a (48 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(48*A + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,37,0.1351,1,"{3046, 2981, 2770, 2774, 216}"
172,1,169,0,0.3869727,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{a} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(8*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{3046, 2981, 2770, 2774, 216}"
173,1,124,0,0.3138655,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a} (8 A+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (8 A+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(8*A + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,37,0.1081,1,"{3046, 2981, 2774, 216}"
174,1,117,0,0.3198391,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,37,0.1081,1,"{3044, 2981, 2774, 216}"
175,1,116,0,0.3090654,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,37,0.1081,1,"{3044, 2980, 2774, 216}"
176,1,123,0,0.3163088,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a (8 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 a (8 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",3,3,37,0.08108,1,"{3044, 2980, 2771}"
177,1,168,0,0.4080908,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a (24 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 a (24 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,37,0.1081,1,"{3044, 2980, 2772, 2771}"
178,1,213,0,0.4632609,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,37,0.1081,1,"{3044, 2980, 2772, 2771}"
179,1,265,0,0.6954798,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (80 A+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}","\frac{a^2 (80 A+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}",1,"(a^(3/2)*(176*A + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(176*A + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,37,0.1622,1,"{3046, 2976, 2981, 2770, 2774, 216}"
180,1,218,0,0.6262167,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (16 A+13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}","\frac{a^2 (16 A+13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^(3/2)*(112*A + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(112*A + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,37,0.1622,1,"{3046, 2976, 2981, 2770, 2774, 216}"
181,1,171,0,0.5175198,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^{3/2} (24 A+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^{3/2} (24 A+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(3/2)*(24*A + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(24*A + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{3046, 2976, 2981, 2774, 216}"
182,1,175,0,0.5288509,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^{3/2} (8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^(3/2)*(8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(8*A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,37,0.1351,1,"{3044, 2976, 2981, 2774, 216}"
183,1,161,0,0.5169747,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}","-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(3*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{3044, 2975, 2981, 2774, 216}"
184,1,163,0,0.4809424,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",5,5,37,0.1351,1,"{3044, 2975, 2980, 2774, 216}"
185,1,172,0,0.5238741,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{6 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{6 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a^2*(4*A + 5*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(104*A + 175*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (6*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,37,0.1081,1,"{3044, 2975, 2980, 2771}"
186,1,219,0,0.6097004,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^2*(52*A + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,5,37,0.1351,1,"{3044, 2975, 2980, 2772, 2771}"
187,1,266,0,0.6997361,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{33 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{33 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",6,5,37,0.1351,1,"{3044, 2975, 2980, 2772, 2771}"
188,1,312,0,0.9126383,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^3 (136 A+109 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}","\frac{a^3 (136 A+109 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1304*A + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(136*A + 109*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(24*A + 23*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d) + (a*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,37,0.1622,1,"{3046, 2976, 2981, 2770, 2774, 216}"
189,1,265,0,0.7926569,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^3 (1040 A+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^{5/2} (400 A+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}","\frac{a^3 (1040 A+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^{5/2} (400 A+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^(5/2)*(400*A + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(400*A + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,37,0.1622,1,"{3046, 2976, 2981, 2770, 2774, 216}"
190,1,218,0,0.713289,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^{5/2} (304 A+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}","\frac{a^{5/2} (304 A+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^(5/2)*(304*A + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(432*A + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,37,0.1351,1,"{3046, 2976, 2981, 2774, 216}"
191,1,222,0,0.7210116,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{5 a^{5/2} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}","\frac{5 a^{5/2} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"(5*a^(5/2)*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(24*A - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,5,37,0.1351,1,"{3044, 2976, 2981, 2774, 216}"
192,1,218,0,0.7234031,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a^{5/2} (8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}",1,"(a^(5/2)*(8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(56*A - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,37,0.1622,1,"{3044, 2975, 2976, 2981, 2774, 216}"
193,1,210,0,0.725002,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(5*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(64*A + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,5,37,0.1351,1,"{3044, 2975, 2981, 2774, 216}"
194,1,210,0,0.6680053,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(32*A + 49*C)*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",6,5,37,0.1351,1,"{3044, 2975, 2980, 2774, 216}"
195,1,219,0,0.7191431,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 a^3 (8 A+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (584 A+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (8 A+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (584 A+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^3*(8*A + 11*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(584*A + 903*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,37,0.1081,1,"{3044, 2975, 2980, 2771}"
196,1,266,0,0.8030206,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",6,5,37,0.1351,1,"{3044, 2975, 2980, 2772, 2771}"
197,1,313,0,0.9103599,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}",1,"(2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))","A",7,5,37,0.1351,1,"{3044, 2975, 2980, 2772, 2771}"
198,1,226,0,0.7492529,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(8 A+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","-\frac{(8 A+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"-((8*A + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,37,0.1892,1,"{3046, 2983, 2982, 2782, 205, 2774, 216}"
199,1,183,0,0.5641089,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}","\frac{(8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"((8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{3046, 2983, 2982, 2782, 205, 2774, 216}"
200,1,133,0,0.3869988,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"-((C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{3046, 2982, 2782, 205, 2774, 216}"
201,1,135,0,0.3917154,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{3044, 2982, 2782, 205, 2774, 216}"
202,1,136,0,0.3479122,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,37,0.1351,1,"{3044, 2984, 12, 2782, 205}"
203,1,181,0,0.5109702,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (13 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 (13 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,37,0.1351,1,"{3044, 2984, 12, 2782, 205}"
204,1,224,0,0.6891402,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (31 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 (31 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,37,0.1351,1,"{3044, 2984, 12, 2782, 205}"
205,1,245,0,0.7866824,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}","\frac{(8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}",1,"((8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,37,0.1892,1,"{3042, 2983, 2982, 2782, 205, 2774, 216}"
206,1,188,0,0.5704499,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}","\frac{(A+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"(-3*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{3042, 2983, 2982, 2782, 205, 2774, 216}"
207,1,145,0,0.410958,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(3 A-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,6,37,0.1622,1,"{3042, 2982, 2782, 205, 2774, 216}"
208,1,152,0,0.3783401,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(7 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,37,0.1351,1,"{3042, 2984, 12, 2782, 205}"
209,1,201,0,0.5493509,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(11 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(11 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,37,0.1351,1,"{3042, 2984, 12, 2782, 205}"
210,1,248,0,0.7393129,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(15 A+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(13 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(49 A+25 C) \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(13 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(49 A+25 C) \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"-((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((49*A + 25*C)*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,37,0.1351,1,"{3042, 2984, 12, 2782, 205}"
211,1,237,0,0.7930726,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(3 A+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(3 A+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-5*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,8,37,0.2162,1,"{3042, 2977, 2983, 2982, 2782, 205, 2774, 216}"
212,1,192,0,0.5798692,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(5 A-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(5 A-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,7,37,0.1892,1,"{3042, 2977, 2982, 2782, 205, 2774, 216}"
213,1,154,0,0.3988079,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(19 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,37,0.1351,1,"{3042, 2978, 12, 2782, 205}"
214,1,199,0,0.5791133,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(49 A+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{5 (15 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{5 (15 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{3042, 2978, 2984, 12, 2782, 205}"
215,1,246,0,0.7696437,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{5 (19 A+3 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(163 A+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{5 (19 A+3 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(163 A+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,6,37,0.1622,1,"{3042, 2978, 2984, 12, 2782, 205}"
216,1,92,0,0.0909403,"\int \cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\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 ^5(c+d x)}{5 d}-\frac{2 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 ^5(c+d x)}{5 d}-\frac{2 C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}",1,"(3*B*x)/8 + (C*Sin[c + d*x])/d + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*C*Sin[c + d*x]^3)/(3*d) + (C*Sin[c + d*x]^5)/(5*d)","A",7,5,28,0.1786,1,"{3010, 2748, 2635, 8, 2633}"
217,1,76,0,0.0845783,"\int \cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{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}","-\frac{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*x)/8 + (B*Sin[c + d*x])/d + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (B*Sin[c + d*x]^3)/(3*d)","A",7,5,28,0.1786,1,"{3010, 2748, 2633, 2635, 8}"
218,1,54,0,0.0631004,"\int \cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}",1,"(B*x)/2 + (C*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (C*Sin[c + d*x]^3)/(3*d)","A",6,5,26,0.1923,1,"{3010, 2748, 2635, 8, 2633}"
219,1,38,0,0.023467,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]","\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}","\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*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,3,19,0.1579,1,"{2637, 2635, 8}"
220,1,15,0,0.0296575,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","B x+\frac{C \sin (c+d x)}{d}","B x+\frac{C \sin (c+d x)}{d}",1,"B*x + (C*Sin[c + d*x])/d","A",3,2,26,0.07692,1,"{3010, 2637}"
221,1,16,0,0.0475166,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x",1,"C*x + (B*ArcTanh[Sin[c + d*x]])/d","A",3,3,28,0.1071,1,"{3010, 2735, 3770}"
222,1,24,0,0.0623378,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(C*ArcTanh[Sin[c + d*x]])/d + (B*Tan[c + d*x])/d","A",5,5,28,0.1786,1,"{3010, 2748, 3767, 8, 3770}"
223,1,47,0,0.075895,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,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 \tan (c+d x)}{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)}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Tan[c + d*x])/d + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,28,0.2143,1,"{3010, 2748, 3768, 3770, 3767, 8}"
224,1,63,0,0.0800781,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{B \tan ^3(c+d x)}{3 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{B \tan ^3(c+d x)}{3 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) + (B*Tan[c + d*x]^3)/(3*d)","A",6,5,28,0.1786,1,"{3010, 2748, 3767, 3768, 3770}"
225,1,85,0,0.0913065,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\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 ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{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 ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d}",1,"(3*B*ArcTanh[Sin[c + d*x]])/(8*d) + (C*Tan[c + d*x])/d + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (C*Tan[c + d*x]^3)/(3*d)","A",7,5,28,0.1786,1,"{3010, 2748, 3768, 3770, 3767}"
226,1,125,0,0.2189209,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a (5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (B+C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (B+C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}","-\frac{a (5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (B+C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (B+C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(3*a*(B + C)*x)/8 + (a*(5*B + 4*C)*Sin[c + d*x])/(5*d) + (3*a*(B + C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*B + 4*C)*Sin[c + d*x]^3)/(15*d)","A",9,7,38,0.1842,1,"{3029, 2968, 3023, 2748, 2633, 2635, 8}"
227,1,97,0,0.1820494,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a (B+C) \sin ^3(c+d x)}{3 d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 B+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}","-\frac{a (B+C) \sin ^3(c+d x)}{3 d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 B+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(4*B + 3*C)*x)/8 + (a*(B + C)*Sin[c + d*x])/d + (a*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(B + C)*Sin[c + d*x]^3)/(3*d)","A",8,7,36,0.1944,1,"{3029, 2968, 3023, 2748, 2635, 8, 2633}"
228,1,85,0,0.0774457,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (3 B+C) \sin (c+d x)}{3 d}+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a x (B+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}","\frac{a (3 B+C) \sin (c+d x)}{3 d}+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a x (B+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}",1,"(a*(B + C)*x)/2 + (a*(3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)","A",2,2,30,0.06667,1,"{3023, 2734}"
229,1,47,0,0.0631725,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a (B+C) \sin (c+d x)}{d}+\frac{1}{2} a x (2 B+C)+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a (B+C) \sin (c+d x)}{d}+\frac{1}{2} a x (2 B+C)+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*B + C)*x)/2 + (a*(B + C)*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",2,2,36,0.05556,1,"{3029, 2734}"
230,1,32,0,0.1452779,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}",1,"a*(B + C)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d","A",5,5,38,0.1316,1,"{3029, 2968, 3023, 2735, 3770}"
231,1,32,0,0.1561561,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+a C x","\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+a C x",1,"a*C*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",5,5,38,0.1316,1,"{3029, 2968, 3021, 2735, 3770}"
232,1,56,0,0.1865268,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*(B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,38,0.1842,1,"{3029, 2968, 3021, 2748, 3767, 8, 3770}"
233,1,86,0,0.2096613,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*B + 3*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",8,8,38,0.2105,1,"{3029, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
234,1,106,0,0.2162961,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 B+4 C) \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 (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(3*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)","A",8,7,38,0.1842,1,"{3029, 2968, 3021, 2748, 3767, 3768, 3770}"
235,1,160,0,0.3388619,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (5 B+6 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 B+6 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}","-\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (5 B+6 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 B+6 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(a^2*(7*B + 6*C)*x)/8 + (a^2*(10*B + 9*C)*Sin[c + d*x])/(5*d) + (a^2*(7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*B + 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(10*B + 9*C)*Sin[c + d*x]^3)/(15*d)","A",9,8,38,0.2105,1,"{3029, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
236,1,129,0,0.1405732,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}","\frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(8*B + 7*C)*x)/8 + (a^2*(8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)","A",3,3,32,0.09375,1,"{3023, 2751, 2644}"
237,1,94,0,0.1246779,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a^2 (3 B+2 C) \sin (c+d x)}{3 d}+\frac{a^2 (3 B+2 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 B+2 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{2 a^2 (3 B+2 C) \sin (c+d x)}{3 d}+\frac{a^2 (3 B+2 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 B+2 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(3*B + 2*C)*x)/2 + (2*a^2*(3*B + 2*C)*Sin[c + d*x])/(3*d) + (a^2*(3*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",3,3,38,0.07895,1,"{3029, 2751, 2644}"
238,1,82,0,0.2725427,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (4 B+3 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}","\frac{a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (4 B+3 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}",1,"(a^2*(4*B + 3*C)*x)/2 + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*B + 3*C)*Sin[c + d*x])/(2*d) + (C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d)","A",6,6,40,0.1500,1,"{3029, 2976, 2968, 3023, 2735, 3770}"
239,1,74,0,0.290653,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (2 B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)","-\frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (2 B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)",1,"a^2*(B + 2*C)*x + (a^2*(2*B + C)*ArcTanh[Sin[c + d*x]])/d - (a^2*(B - C)*Sin[c + d*x])/d + (B*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d","A",6,6,40,0.1500,1,"{3029, 2975, 2968, 3023, 2735, 3770}"
240,1,88,0,0.2997054,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+a^2 C x","\frac{a^2 (3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+a^2 C x",1,"a^2*C*x + (a^2*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*B + 2*C)*Tan[c + d*x])/(2*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,40,0.1500,1,"{3029, 2975, 2968, 3021, 2735, 3770}"
241,1,113,0,0.3598728,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (2 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 B+3 C) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{B \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}","\frac{a^2 (5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (2 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 B+3 C) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{B \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}",1,"(a^2*(2*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(5*B + 6*C)*Tan[c + d*x])/(3*d) + (a^2*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",8,8,40,0.2000,1,"{3029, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
242,1,144,0,0.3862323,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (4 B+5 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{4 d}","\frac{a^2 (4 B+5 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{4 d}",1,"(a^2*(7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*B + 5*C)*Tan[c + d*x])/(3*d) + (a^2*(7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",9,9,40,0.2250,1,"{3029, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
243,1,169,0,0.3935335,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (6 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}","\frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (6 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(a^2*(6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(9*B + 10*C)*Tan[c + d*x])/(5*d) + (a^2*(6*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(6*B + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a^2*(9*B + 10*C)*Tan[c + d*x]^3)/(15*d)","A",9,8,40,0.2000,1,"{3029, 2975, 2968, 3021, 2748, 3767, 3768, 3770}"
244,1,201,0,0.4868704,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^3 (19 B+17 C) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 B+17 C) \sin (c+d x)}{5 d}+\frac{a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 B+23 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 B+23 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}","-\frac{a^3 (19 B+17 C) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 B+17 C) \sin (c+d x)}{5 d}+\frac{a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 B+23 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 B+23 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^3*(26*B + 23*C)*x)/16 + (a^3*(19*B + 17*C)*Sin[c + d*x])/(5*d) + (a^3*(26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(22*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*B + 4*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(19*B + 17*C)*Sin[c + d*x]^3)/(15*d)","A",10,8,38,0.2105,1,"{3029, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
245,1,154,0,0.1940154,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}","-\frac{a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(15*B + 13*C)*x)/8 + (a^3*(15*B + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(15*B + 13*C)*Sin[c + d*x]^3)/(60*d)","A",9,7,32,0.2188,1,"{3023, 2751, 2645, 2637, 2635, 8, 2633}"
246,1,116,0,0.166009,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac{3 a^3 (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 B+3 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","-\frac{a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac{3 a^3 (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 B+3 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(5*a^3*(4*B + 3*C)*x)/8 + (a^3*(4*B + 3*C)*Sin[c + d*x])/d + (3*a^3*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(4*B + 3*C)*Sin[c + d*x]^3)/(12*d)","A",9,7,38,0.1842,1,"{3029, 2751, 2645, 2637, 2635, 8, 2633}"
247,1,111,0,0.3813816,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(3 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (7 B+5 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(3 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (7 B+5 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^3*(7*B + 5*C)*x)/2 + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) + (a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d)","A",7,6,40,0.1500,1,"{3029, 2976, 2968, 3023, 2735, 3770}"
248,1,110,0,0.4003053,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 B+7 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a \cos (c+d x)+a)^2}{d}","\frac{a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 B+7 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^3*(6*B + 7*C)*x)/2 + (a^3*(3*B + C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - ((2*B - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d","A",7,7,40,0.1750,1,"{3029, 2975, 2976, 2968, 3023, 2735, 3770}"
249,1,114,0,0.4254656,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 B+C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}","\frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 B+C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"a^3*(B + 3*C)*x + (a^3*(7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*B*Sin[c + d*x])/(2*d) + ((2*B + C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/d + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,40,0.1500,1,"{3029, 2975, 2968, 3023, 2735, 3770}"
250,1,125,0,0.4235763,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{a^3 (5 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 B+3 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 C x+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{a^3 (5 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 B+3 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 C x+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"a^3*C*x + (a^3*(5*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) + ((5*B + 3*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,6,40,0.1500,1,"{3029, 2975, 2968, 3021, 2735, 3770}"
251,1,154,0,0.5055964,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^3 (9 B+11 C) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 B+28 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 B+2 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}","\frac{a^3 (9 B+11 C) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 B+28 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 B+2 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"(5*a^3*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(9*B + 11*C)*Tan[c + d*x])/(3*d) + (a^3*(27*B + 28*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*B + 2*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",9,8,40,0.2000,1,"{3029, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
252,1,185,0,0.5334252,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^3 (38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 B+15 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{a^3 (38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 B+15 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^3*(13*B + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(13*B + 15*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*B + 45*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((7*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",10,9,40,0.2250,1,"{3029, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
253,1,122,0,0.26259,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{(3 B-4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (B-C)}{2 a}","\frac{(3 B-4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (B-C)}{2 a}",1,"(3*(B - C)*x)/(2*a) - ((3*B - 4*C)*Sin[c + d*x])/(a*d) + (3*(B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*B - 4*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,6,40,0.1500,1,"{3029, 2977, 2748, 2635, 8, 2633}"
254,1,99,0,0.1709558,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 B-3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 B-3 C)}{2 a}","\frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 B-3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 B-3 C)}{2 a}",1,"-((2*B - 3*C)*x)/(2*a) + (2*(B - C)*Sin[c + d*x])/(a*d) - ((2*B - 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",3,3,38,0.07895,1,"{3029, 2977, 2734}"
255,1,54,0,0.088818,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","-\frac{(B-C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}","-\frac{(B-C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}",1,"((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) - ((B - C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))","A",4,4,32,0.1250,1,"{3023, 12, 2735, 2648}"
256,1,34,0,0.1213685,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{C x}{a}","\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{C x}{a}",1,"(C*x)/a + ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",3,3,38,0.07895,1,"{3029, 2735, 2648}"
257,1,44,0,0.1659255,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(B*ArcTanh[Sin[c + d*x]])/(a*d) - ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,40,0.1000,1,"{3029, 2978, 12, 3770}"
258,1,69,0,0.239387,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","\frac{(2 B-C) \tan (c+d x)}{a d}-\frac{(B-C) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \tan (c+d x)}{d (a \cos (c+d x)+a)}","\frac{(2 B-C) \tan (c+d x)}{a d}-\frac{(B-C) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"-(((B - C)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*B - C)*Tan[c + d*x])/(a*d) - ((B - C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,40,0.1500,1,"{3029, 2978, 2748, 3767, 8, 3770}"
259,1,107,0,0.2486586,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","-\frac{2 (B-C) \tan (c+d x)}{a d}+\frac{(3 B-2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 B-2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}","-\frac{2 (B-C) \tan (c+d x)}{a d}+\frac{(3 B-2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 B-2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"((3*B - 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(B - C)*Tan[c + d*x])/(a*d) + ((3*B - 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,7,40,0.1750,1,"{3029, 2978, 2748, 3768, 3770, 3767, 8}"
260,1,131,0,0.2647445,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + a*Cos[c + d*x]),x]","\frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}","\frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(-3*(B - C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((4*B - 3*C)*Tan[c + d*x])/(a*d) - (3*(B - C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*B - 3*C)*Tan[c + d*x]^3)/(3*a*d)","A",7,6,40,0.1500,1,"{3029, 2978, 2748, 3767, 3768, 3770}"
261,1,170,0,0.3967879,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{4 (2 B-3 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 B-3 C) \sin (c+d x)}{a^2 d}+\frac{(7 B-10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 B-10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 B-10 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{4 (2 B-3 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 B-3 C) \sin (c+d x)}{a^2 d}+\frac{(7 B-10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 B-10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 B-10 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*B - 10*C)*x)/(2*a^2) - (4*(2*B - 3*C)*Sin[c + d*x])/(a^2*d) + ((7*B - 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((7*B - 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(2*B - 3*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",8,6,40,0.1500,1,"{3029, 2977, 2748, 2635, 8, 2633}"
262,1,147,0,0.3625895,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}+\frac{(5 B-8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 B-7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 B-7 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}+\frac{(5 B-8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 B-7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 B-7 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-((4*B - 7*C)*x)/(2*a^2) + (2*(5*B - 8*C)*Sin[c + d*x])/(3*a^2*d) - ((4*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((5*B - 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",4,3,40,0.07500,1,"{3029, 2977, 2734}"
263,1,99,0,0.3149391,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(B-4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(B-4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((B - 2*C)*x)/a^2 - ((B - 4*C)*Sin[c + d*x])/(3*a^2*d) - ((B - 2*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,38,0.1842,1,"{3029, 2977, 2968, 3023, 12, 2735, 2648}"
264,1,70,0,0.1106964,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{(2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^2 + ((2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,32,0.09375,1,"{3019, 2735, 2648}"
265,1,65,0,0.1240322,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","\frac{(B+2 C) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(B+2 C) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((B + 2*C)*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))","A",3,3,38,0.07895,1,"{3029, 2750, 2648}"
266,1,79,0,0.2657888,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","-\frac{(4 B-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(4 B-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(B*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((4*B - C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,4,40,0.1000,1,"{3029, 2978, 12, 3770}"
267,1,107,0,0.3797486,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","\frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((2*B - C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(5*B - 2*C)*Tan[c + d*x])/(3*a^2*d) - ((2*B - C)*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,40,0.1500,1,"{3029, 2978, 2748, 3767, 8, 3770}"
268,1,152,0,0.4009426,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 B-4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 B-5 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 B-4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 B-5 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*B - 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*B - 5*C)*Tan[c + d*x])/(3*a^2*d) + ((7*B - 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*B - 5*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,7,40,0.1750,1,"{3029, 2978, 2748, 3768, 3770, 3767, 8}"
269,1,193,0,0.5345341,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{8 (9 B-19 C) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 B-19 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 B-13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 B-13 C)}{2 a^3}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 B-11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{8 (9 B-19 C) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 B-19 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 B-13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 B-13 C)}{2 a^3}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 B-11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((6*B - 13*C)*x)/(2*a^3) + (8*(9*B - 19*C)*Sin[c + d*x])/(15*a^3*d) - ((6*B - 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*B - 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*B - 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,3,40,0.07500,1,"{3029, 2977, 2734}"
270,1,147,0,0.5148696,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{(7 B-27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{(7 B-27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((B - 3*C)*x)/a^3 - ((7*B - 27*C)*Sin[c + d*x])/(15*a^3*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,40,0.1750,1,"{3029, 2977, 2968, 3023, 12, 2735, 2648}"
271,1,116,0,0.3381887,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^3 + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*B - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((4*B - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",6,6,38,0.1579,1,"{3029, 2977, 2968, 3019, 2735, 2648}"
272,1,102,0,0.1341523,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{(3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*B + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,32,0.09375,1,"{3019, 2750, 2648}"
273,1,102,0,0.1540598,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","\frac{(2 B+3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 B+3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(2 B+3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 B+3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*B + 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*B + 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,4,38,0.1053,1,"{3029, 2750, 2650, 2648}"
274,1,117,0,0.402083,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (11 B-C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 B-2 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{2 (11 B-C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 B-2 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(B*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*B - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*B - C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",6,4,40,0.1000,1,"{3029, 2978, 12, 3770}"
275,1,145,0,0.5460173,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","\frac{2 (36 B-11 C) \tan (c+d x)}{15 a^3 d}-\frac{(3 B-C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 B-C) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 B-4 C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{2 (36 B-11 C) \tan (c+d x)}{15 a^3 d}-\frac{(3 B-C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 B-C) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 B-4 C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-(((3*B - C)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*B - 11*C)*Tan[c + d*x])/(15*a^3*d) - ((B - C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*B - 4*C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*B - C)*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,40,0.1500,1,"{3029, 2978, 2748, 3767, 8, 3770}"
276,1,196,0,0.5657935,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","-\frac{8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 B-6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 B-9 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 B-6 C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 B-6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 B-9 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 B-6 C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((13*B - 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*B - 9*C)*Tan[c + d*x])/(15*a^3*d) + ((13*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*B - 9*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",9,7,40,0.1750,1,"{3029, 2978, 2748, 3768, 3770, 3767, 8}"
277,1,101,0,0.1323357,"\int \sqrt{a+a \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}","\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(2*a*(5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",3,3,34,0.08824,1,"{3023, 2751, 2646}"
278,1,138,0,0.1882778,"\int (a+a \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}","\frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}",1,"(8*a^2*(21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",4,4,34,0.1176,1,"{3023, 2751, 2647, 2646}"
279,1,175,0,0.2225639,"\int (a+a \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{16 a^2 (15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}","\frac{16 a^2 (15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}",1,"(64*a^3*(15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)","A",5,4,34,0.1176,1,"{3023, 2751, 2647, 2646}"
280,1,118,0,0.1512497,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}","\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"-((Sqrt[2]*(B - C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",4,4,34,0.1176,1,"{3023, 2751, 2649, 206}"
281,1,118,0,0.1517329,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(B-C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}","\frac{(3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(B-C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((3*B - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,34,0.1176,1,"{3019, 2751, 2649, 206}"
282,1,126,0,0.1719193,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(B-C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(B-C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((5*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,34,0.1176,1,"{3019, 2750, 2649, 206}"
283,1,111,0,0.1071445,"\int \cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 C \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 C \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*C*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,5,30,0.1667,1,"{3010, 2748, 2635, 2639, 2641}"
284,1,87,0,0.0881781,"\int \sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(6*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,5,30,0.1667,1,"{3010, 2748, 2635, 2641, 2639}"
285,1,61,0,0.0786756,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d + (2*C*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,30,0.1667,1,"{3010, 2748, 2639, 2635, 2641}"
286,1,35,0,0.0662762,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*C*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d","A",4,4,30,0.1333,1,"{3010, 2748, 2641, 2639}"
287,1,57,0,0.0746286,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*C*EllipticF[(c + d*x)/2, 2])/d + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,30,0.1667,1,"{3010, 2748, 2636, 2639, 2641}"
288,1,83,0,0.0843538,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*C*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,5,30,0.1667,1,"{3010, 2748, 2636, 2641, 2639}"
289,1,111,0,0.0988952,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2),x]","-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,5,30,0.1667,1,"{3010, 2748, 2636, 2639, 2641}"
290,1,132,0,0.1140003,"\int \cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(6 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(6 A+5 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 A+5 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}+\frac{C \sin (c+d x) \cos ^5(c+d x)}{6 d}","\frac{(6 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(6 A+5 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 A+5 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}+\frac{C \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"((6*A + 5*C)*x)/16 + (B*Sin[c + d*x])/d + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d)","A",7,5,29,0.1724,1,"{3023, 2748, 2635, 8, 2633}"
291,1,113,0,0.1072832,"\int \cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \sin (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}+\frac{C \sin (c+d x) \cos ^4(c+d x)}{5 d}","-\frac{(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \sin (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}+\frac{C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(3*B*x)/8 + ((5*A + 4*C)*Sin[c + d*x])/(5*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A + 4*C)*Sin[c + d*x]^3)/(15*d)","A",7,5,29,0.1724,1,"{3023, 2748, 2633, 2635, 8}"
292,1,88,0,0.0945887,"\int \cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 A+3 C)-\frac{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{(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 A+3 C)-\frac{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}",1,"((4*A + 3*C)*x)/8 + (B*Sin[c + d*x])/d + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (B*Sin[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3023, 2748, 2635, 8, 2633}"
293,1,69,0,0.0470357,"\int \cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(3 A+2 C) \sin (c+d x)}{3 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 d}","\frac{(3 A+2 C) \sin (c+d x)}{3 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(B*x)/2 + ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",2,2,27,0.07407,1,"{3023, 2734}"
294,1,41,0,0.0246001,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[A + B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]","A x+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}","A x+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}",1,"A*x + (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,3,20,0.1500,1,"{2637, 2635, 8}"
295,1,27,0,0.0521309,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c+d x)}{d}","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c+d x)}{d}",1,"B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d","A",3,3,27,0.1111,1,"{3023, 2735, 3770}"
296,1,27,0,0.0540409,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{A \tan (c+d x)}{d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x","\frac{A \tan (c+d x)}{d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x",1,"C*x + (B*ArcTanh[Sin[c + d*x]])/d + (A*Tan[c + d*x])/d","A",3,3,29,0.1034,1,"{3021, 2735, 3770}"
297,1,51,0,0.0790041,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{B \tan (c+d x)}{d}","\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{B \tan (c+d x)}{d}",1,"((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,29,0.1724,1,"{3021, 2748, 3767, 8, 3770}"
298,1,78,0,0.1026897,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{(2 A+3 C) \tan (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(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{(2 A+3 C) \tan (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(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}",1,"(B*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A + 3*C)*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{3021, 2748, 3768, 3770, 3767, 8}"
299,1,97,0,0.1038974,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}","\frac{(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}",1,"((3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (B*Tan[c + d*x])/d + ((3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (B*Tan[c + d*x]^3)/(3*d)","A",6,5,29,0.1724,1,"{3021, 2748, 3767, 3768, 3770}"
300,1,122,0,0.1183194,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(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{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(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}",1,"(3*B*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*A + 5*C)*Tan[c + d*x])/(5*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*A + 5*C)*Tan[c + d*x]^3)/(15*d)","A",7,5,29,0.1724,1,"{3021, 2748, 3768, 3770, 3767}"
301,1,143,0,0.2288793,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 (B+C))+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}","-\frac{a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 (B+C))+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(4*A + 3*(B + C))*x)/8 + (a*(5*A + 5*B + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*(B + C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 5*B + 4*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{3033, 3023, 2748, 2635, 8, 2633}"
302,1,118,0,0.1381483,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+4 B+3 C)+\frac{a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}","\frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+4 B+3 C)+\frac{a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(4*A + 4*B + 3*C)*x)/8 + (a*(3*A + 2*(B + C))*Sin[c + d*x])/(3*d) + (a*(4*A + 4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",3,3,37,0.08108,1,"{3033, 3023, 2734}"
303,1,91,0,0.0831877,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+B+C)+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}","\frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+B+C)+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}",1,"(a*(2*A + B + C)*x)/2 + (a*(3*A + 3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)","A",2,2,31,0.06452,1,"{3023, 2734}"
304,1,63,0,0.135698,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{1}{2} a x (2 A+2 B+C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{1}{2} a x (2 A+2 B+C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*A + 2*B + C)*x)/2 + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*(B + C)*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,37,0.1081,1,"{3033, 3023, 2735, 3770}"
305,1,46,0,0.1361313,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}",1,"a*(B + C)*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d","A",4,4,39,0.1026,1,"{3031, 3023, 2735, 3770}"
306,1,62,0,0.1577487,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x","\frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x",1,"a*C*x + (a*(A + 2*(B + C))*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,39,0.1026,1,"{3031, 3021, 2735, 3770}"
307,1,91,0,0.2112657,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (2 A+3 (B+C)) \tan (c+d x)}{3 d}+\frac{a (A+B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{a (2 A+3 (B+C)) \tan (c+d x)}{3 d}+\frac{a (A+B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(A + B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*(B + C))*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,39,0.1538,1,"{3031, 3021, 2748, 3767, 8, 3770}"
308,1,125,0,0.2538428,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{a (-3 (A+B+C)+A+B) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}","-\frac{a (-3 (A+B+C)+A+B) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*A + 4*(B + C))*ArcTanh[Sin[c + d*x]])/(8*d) - (a*(A + B - 3*(A + B + C))*Tan[c + d*x])/(3*d) + (a*(3*A + 4*(B + C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,39,0.1795,1,"{3031, 3021, 2748, 3768, 3770, 3767, 8}"
309,1,213,0,0.5030672,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+12 B+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+12 B+11 C)+\frac{(3 B+C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}","-\frac{a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+12 B+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+12 B+11 C)+\frac{(3 B+C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^2*(14*A + 12*B + 11*C)*x)/16 + (a^2*(10*A + 9*B + 8*C)*Sin[c + d*x])/(5*d) + (a^2*(14*A + 12*B + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(10*A + 12*B + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*B + C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^2*(10*A + 9*B + 8*C)*Sin[c + d*x]^3)/(15*d)","A",9,8,41,0.1951,1,"{3045, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
310,1,181,0,0.3356722,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B+6 C)+\frac{(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B+6 C)+\frac{(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(8*A + 7*B + 6*C)*x)/8 + (a^2*(8*A + 7*B + 6*C)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((20*A - 5*B + 6*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + ((5*B + 2*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*a*d)","A",5,5,39,0.1282,1,"{3045, 2968, 3023, 2751, 2644}"
311,1,138,0,0.1747887,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (12 A+8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}","\frac{a^2 (12 A+8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(12*A + 8*B + 7*C)*x)/8 + (a^2*(12*A + 8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)","A",3,3,33,0.09091,1,"{3023, 2751, 2644}"
312,1,120,0,0.3433513,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B+2 C)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B+2 C)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(4*A + 3*B + 2*C)*x)/2 + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B + 2*C)*Sin[c + d*x])/(2*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(6*d)","A",6,6,39,0.1538,1,"{3045, 2976, 2968, 3023, 2735, 3770}"
313,1,121,0,0.3838191,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+4 B+3 C)-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}","-\frac{a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+4 B+3 C)-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^2*(2*A + 4*B + 3*C)*x)/2 + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 2*B - 3*C)*Sin[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d","A",6,6,41,0.1463,1,"{3043, 2976, 2968, 3023, 2735, 3770}"
314,1,123,0,0.3924615,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A+B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}","-\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A+B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"a^2*(B + 2*C)*x + (a^2*(3*A + 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A + 2*B - 2*C)*Sin[c + d*x])/(2*d) + ((A + B)*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,41,0.1463,1,"{3043, 2975, 2968, 3023, 2735, 3770}"
315,1,134,0,0.3851802,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (2 A+3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (2 A+3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{a^2 (2 A+3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (2 A+3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"a^2*C*x + (a^2*(2*A + 3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B + 2*C)*Tan[c + d*x])/(2*d) + ((2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{3043, 2975, 2968, 3021, 2735, 3770}"
316,1,160,0,0.4745547,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (4 A+5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+16 B+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}","\frac{a^2 (4 A+5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+16 B+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"(a^2*(7*A + 8*B + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*A + 5*B + 6*C)*Tan[c + d*x])/(3*d) + (a^2*(11*A + 16*B + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((A + 2*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,8,41,0.1951,1,"{3043, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
317,1,196,0,0.5126524,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (18 A+20 B+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (6 A+7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (18 A+25 B+20 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^2 (6 A+7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(2 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{a^2 (18 A+20 B+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (6 A+7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (18 A+25 B+20 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^2 (6 A+7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(2 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(6*A + 7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(18*A + 20*B + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(6*A + 7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(18*A + 25*B + 20*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((2*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,9,41,0.2195,1,"{3043, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
318,1,265,0,0.678233,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B+21 C)+\frac{(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}","-\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B+21 C)+\frac{(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*(26*A + 23*B + 21*C)*x)/16 + (a^3*(133*A + 119*B + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 23*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(154*A + 147*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((7*B + 3*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(133*A + 119*B + 108*C)*Sin[c + d*x]^3)/(105*d)","A",10,8,41,0.1951,1,"{3045, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
319,1,207,0,0.3962152,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+26 B+23 C)+\frac{(30 A-6 B+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}","-\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+26 B+23 C)+\frac{(30 A-6 B+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"(a^3*(30*A + 26*B + 23*C)*x)/16 + (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((2*B + C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d) - (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x]^3)/(120*d)","A",11,9,39,0.2308,1,"{3045, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
320,1,166,0,0.2263782,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}","-\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(20*A + 15*B + 13*C)*x)/8 + (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x]^3)/(60*d)","A",9,7,33,0.2121,1,"{3023, 2751, 2645, 2637, 2635, 8, 2633}"
321,1,162,0,0.4779429,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac{(12 A+20 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+\frac{1}{8} a^3 x (28 A+20 B+15 C)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 B+3 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac{(12 A+20 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+\frac{1}{8} a^3 x (28 A+20 B+15 C)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 B+3 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(28*A + 20*B + 15*C)*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 4*B + 3*C)*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((4*B + 3*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*a*d) + ((12*A + 20*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(24*d)","A",7,6,39,0.1538,1,"{3045, 2976, 2968, 3023, 2735, 3770}"
322,1,156,0,0.5095207,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{(6 A-3 B-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+7 B+5 C)-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}","-\frac{(6 A-3 B-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+7 B+5 C)-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^3*(6*A + 7*B + 5*C)*x)/2 + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d) - ((6*A - 3*B - 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d","A",7,6,41,0.1463,1,"{3043, 2976, 2968, 3023, 2735, 3770}"
323,1,175,0,0.5335268,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A+2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{(3 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (2 A+6 B+7 C)-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}","\frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A+2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{(3 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (2 A+6 B+7 C)-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^3*(2*A + 6*B + 7*C)*x)/2 + (a^3*(7*A + 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - ((4*A + 2*B - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + ((3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,41,0.1707,1,"{3043, 2975, 2976, 2968, 3023, 2735, 3770}"
324,1,169,0,0.5718309,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^3 (5 A+7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+6 B+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+a^3 x (B+3 C)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}","\frac{a^3 (5 A+7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+6 B+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+a^3 x (B+3 C)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"a^3*(B + 3*C)*x + (a^3*(5*A + 7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + ((5*A + 6*B + 3*C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/(3*d) + ((A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,6,41,0.1463,1,"{3043, 2975, 2968, 3023, 2735, 3770}"
325,1,183,0,0.5537341,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{5 a^3 (3 A+4 (B+C)) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+20 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(15 A+20 B+12 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+\frac{(3 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+a^3 C x+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^3 (3 A+4 (B+C)) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+20 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(15 A+20 B+12 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+\frac{(3 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+a^3 C x+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"a^3*C*x + (a^3*(15*A + 20*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*(B + C))*Tan[c + d*x])/(8*d) + ((15*A + 20*B + 12*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A + 4*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,6,41,0.1463,1,"{3043, 2975, 2968, 3021, 2735, 3770}"
326,1,212,0,0.6370812,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^3 (38 A+45 B+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+135 B+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+15 B+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{(3 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}","\frac{a^3 (38 A+45 B+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+135 B+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+15 B+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{(3 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(a^3*(13*A + 15*B + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 45*B + 55*C)*Tan[c + d*x])/(15*d) + (a^3*(109*A + 135*B + 140*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((11*A + 15*B + 10*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((3*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,8,41,0.1951,1,"{3043, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
327,1,244,0,0.7034894,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{(A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}","\frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{(A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"(a^3*(23*A + 26*B + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 26*B + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 86*B + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 42*B + 30*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",10,9,41,0.2195,1,"{3043, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
328,1,304,0,0.8626274,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{7 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+352 B+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+352 B+323 C)+\frac{a (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}","-\frac{a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{(56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{7 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+352 B+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+352 B+323 C)+\frac{a (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}",1,"(a^4*(392*A + 352*B + 323*C)*x)/128 + (a^4*(252*A + 227*B + 208*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 352*B + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2408*A + 2208*B + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*(2*B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((56*A + 80*B + 61*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(8*A + 8*B + 7*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(252*A + 227*B + 208*C)*Sin[c + d*x]^3)/(105*d)","A",11,8,41,0.1951,1,"{3045, 2976, 2968, 3023, 2748, 2635, 8, 2633}"
329,1,243,0,0.4544721,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac{1}{16} a^4 x (56 A+49 B+44 C)+\frac{(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}","-\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac{1}{16} a^4 x (56 A+49 B+44 C)+\frac{(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}",1,"(a^4*(56*A + 49*B + 44*C)*x)/16 + (4*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]*Sin[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(210*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(42*a*d) - (2*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x]^3)/(105*d)","A",14,9,39,0.2308,1,"{3045, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
330,1,200,0,0.2796672,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+8 B+7 C)+\frac{(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}","-\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+8 B+7 C)+\frac{(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}",1,"(7*a^4*(10*A + 8*B + 7*C)*x)/16 + (4*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*B - C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x]^3)/(15*d)","A",12,7,33,0.2121,1,"{3023, 2751, 2645, 2637, 2635, 8, 2633}"
331,1,195,0,0.6030967,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac{(20 A+35 B+28 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{(32 A+35 B+28 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (48 A+35 B+28 C)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac{(20 A+35 B+28 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{(32 A+35 B+28 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (48 A+35 B+28 C)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*(48*A + 35*B + 28*C)*x)/8 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(40*A + 35*B + 28*C)*Sin[c + d*x])/(8*d) + (a*(5*B + 4*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d)","A",8,6,39,0.1538,1,"{3045, 2976, 2968, 3023, 2735, 3770}"
332,1,196,0,0.6791152,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-4 B-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{(12 A-32 B-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+48 B+35 C)-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}","\frac{5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-4 B-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{(12 A-32 B-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+48 B+35 C)-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}",1,"(a^4*(52*A + 48*B + 35*C)*x)/8 + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 8*B + 7*C)*Sin[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - ((12*A - 4*B - 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - ((12*A - 32*B - 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d","A",8,6,41,0.1463,1,"{3043, 2976, 2968, 3023, 2735, 3770}"
333,1,206,0,0.6855337,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A+6 B-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}-\frac{(18 A+3 B-8 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (8 A+13 B+12 C)+\frac{a (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}","-\frac{5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A+6 B-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}-\frac{(18 A+3 B-8 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (8 A+13 B+12 C)+\frac{a (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}",1,"(a^4*(8*A + 13*B + 12*C)*x)/2 + (a^4*(13*A + 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B - 2*C)*Sin[c + d*x])/(2*d) - ((15*A + 6*B - 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((18*A + 3*B - 8*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*(2*A + B)*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",8,7,41,0.1707,1,"{3043, 2975, 2976, 2968, 3023, 2735, 3770}"
334,1,219,0,0.7137351,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(22 A+18 B+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{(16 A+15 B+6 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}+\frac{1}{2} a^4 x (2 A+8 B+13 C)+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}","-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(22 A+18 B+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{(16 A+15 B+6 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}+\frac{1}{2} a^4 x (2 A+8 B+13 C)+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}",1,"(a^4*(2*A + 8*B + 13*C)*x)/2 + (a^4*(12*A + 13*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B - C)*Sin[c + d*x])/(2*d) - ((22*A + 18*B + 3*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + ((16*A + 15*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + (a*(4*A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",8,7,41,0.1707,1,"{3043, 2975, 2976, 2968, 3023, 2735, 3770}"
335,1,217,0,0.7427335,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+44 B+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+\frac{(7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+a^4 x (B+4 C)+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}","-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+44 B+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+\frac{(7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+a^4 x (B+4 C)+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}",1,"a^4*(B + 4*C)*x + (a^4*(35*A + 48*B + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 8*B + 4*C)*Sin[c + d*x])/(8*d) + ((35*A + 44*B + 36*C)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(12*d) + ((7*A + 8*B + 4*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,6,41,0.1463,1,"{3043, 2975, 2968, 3023, 2735, 3770}"
336,1,225,0,0.6910088,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac{a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(28 A+35 B+20 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{(28 A+35 B+32 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 C x+\frac{a (4 A+5 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac{a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(28 A+35 B+20 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{(28 A+35 B+32 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 C x+\frac{a (4 A+5 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"a^4*C*x + (a^4*(28*A + 35*B + 48*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(28*A + 35*B + 40*C)*Tan[c + d*x])/(8*d) + ((28*A + 35*B + 32*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((28*A + 35*B + 20*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + (a*(4*A + 5*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,6,41,0.1463,1,"{3043, 2975, 2968, 3021, 2735, 3770}"
337,1,253,0,0.8334458,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{(43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{a (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}","\frac{a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{(43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{a (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}",1,"(7*a^4*(7*A + 8*B + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B + 100*C)*Tan[c + d*x])/(15*d) + (a^4*(417*A + 488*B + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((43*A + 52*B + 50*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((37*A + 48*B + 30*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*(2*A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",10,8,41,0.1951,1,"{3043, 2975, 2968, 3021, 2748, 3767, 8, 3770}"
338,1,287,0,0.8680609,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{70 d}+\frac{(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{840 d}+\frac{a (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}","\frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{70 d}+\frac{(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{840 d}+\frac{a (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}",1,"(a^4*(44*A + 49*B + 56*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(454*A + 504*B + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Sec[c + d*x]^2*Tan[c + d*x])/(840*d) + ((436*A + 511*B + 504*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(840*d) + ((16*A + 21*B + 14*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(70*d) + (a*(4*A + 7*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)","A",11,9,41,0.2195,1,"{3043, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
339,1,174,0,0.2337502,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A-4 B+5 C)}{8 a}","\frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A-4 B+5 C)}{8 a}",1,"(3*(4*A - 4*B + 5*C)*x)/(8*a) - ((3*A - 4*B + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,5,41,0.1220,1,"{3041, 2748, 2633, 2635, 8}"
340,1,139,0,0.2118742,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B+3 C)}{2 a}","-\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B+3 C)}{2 a}",1,"-((2*A - 3*B + 3*C)*x)/(2*a) + ((3*A - 3*B + 4*C)*Sin[c + d*x])/(a*d) - ((2*A - 3*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - ((3*A - 3*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,41,0.1220,1,"{3041, 2748, 2635, 8, 2633}"
341,1,110,0,0.1225437,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(A-2 B+2 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A-2 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A-2 B+3 C)}{2 a}","-\frac{(A-2 B+2 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A-2 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A-2 B+3 C)}{2 a}",1,"((2*A - 2*B + 3*C)*x)/(2*a) - ((A - 2*B + 2*C)*Sin[c + d*x])/(a*d) + ((2*A - 2*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",2,2,39,0.05128,1,"{3041, 2734}"
342,1,54,0,0.1121866,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{(A-B+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}","\frac{(A-B+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}",1,"((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) + ((A - B + C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))","A",3,3,33,0.09091,1,"{3023, 2735, 2648}"
343,1,51,0,0.1193685,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","-\frac{(A-B+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}","-\frac{(A-B+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}",1,"(C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",3,3,39,0.07692,1,"{3041, 2735, 3770}"
344,1,71,0,0.1732976,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{(2 A-B+C) \tan (c+d x)}{a d}-\frac{(A-B+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}","\frac{(2 A-B+C) \tan (c+d x)}{a d}-\frac{(A-B+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}",1,"-(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B + C)*Tan[c + d*x])/(a*d) - ((A - B + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,41,0.1220,1,"{3041, 2748, 3767, 8, 3770}"
345,1,117,0,0.2016248,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","-\frac{(2 A-2 B+C) \tan (c+d x)}{a d}+\frac{(3 A-2 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}","-\frac{(2 A-2 B+C) \tan (c+d x)}{a d}+\frac{(3 A-2 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"((3*A - 2*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A - 2*B + C)*Tan[c + d*x])/(a*d) + ((3*A - 2*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,41,0.1463,1,"{3041, 2748, 3768, 3770, 3767, 8}"
346,1,148,0,0.2168428,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{(4 A-3 B+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B+3 C) \tan (c+d x)}{a d}-\frac{(3 A-3 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A-3 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}","\frac{(4 A-3 B+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B+3 C) \tan (c+d x)}{a d}-\frac{(3 A-3 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A-3 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"-((3*A - 3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((4*A - 3*B + 3*C)*Tan[c + d*x])/(a*d) - ((3*A - 3*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A - 3*B + 3*C)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,41,0.1220,1,"{3041, 2748, 3767, 3768, 3770}"
347,1,185,0,0.3791381,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B+10 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B+10 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-((4*A - 7*B + 10*C)*x)/(2*a^2) + ((5*A - 8*B + 12*C)*Sin[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 8*B + 12*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,6,41,0.1463,1,"{3041, 2977, 2748, 2635, 8, 2633}"
348,1,160,0,0.3167924,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (2 A-5 B+8 C) \sin (c+d x)}{3 a^2 d}-\frac{(2 A-5 B+8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-4 B+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A-4 B+7 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{2 (2 A-5 B+8 C) \sin (c+d x)}{3 a^2 d}-\frac{(2 A-5 B+8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-4 B+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A-4 B+7 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*A - 4*B + 7*C)*x)/(2*a^2) - (2*(2*A - 5*B + 8*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A - 4*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((2*A - 5*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,41,0.07317,1,"{3041, 2977, 2734}"
349,1,103,0,0.2585338,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((B - 2*C)*x)/a^2 + ((A - B + 4*C)*Sin[c + d*x])/(3*a^2*d) - ((B - 2*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,39,0.1538,1,"{3041, 2968, 3023, 12, 2735, 2648}"
350,1,72,0,0.1197888,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^2 + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,33,0.09091,1,"{3019, 2735, 2648}"
351,1,83,0,0.2148221,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","-\frac{(4 A-B-2 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(4 A-B-2 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((4*A - B - 2*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",4,4,39,0.1026,1,"{3041, 2978, 12, 3770}"
352,1,109,0,0.3448096,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{(10 A-4 B+C) \tan (c+d x)}{3 a^2 d}-\frac{(2 A-B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 A-B) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(10 A-4 B+C) \tan (c+d x)}{3 a^2 d}-\frac{(2 A-B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 A-B) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((2*A - B)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((10*A - 4*B + C)*Tan[c + d*x])/(3*a^2*d) - ((2*A - B)*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 8, 3770}"
353,1,165,0,0.3607678,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (8 A-5 B+2 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B+2 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{2 (8 A-5 B+2 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B+2 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A - 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B + 2*C)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*A - 5*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,41,0.1707,1,"{3041, 2978, 2748, 3768, 3770, 3767, 8}"
354,1,194,0,0.3875338,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-((10*A - 7*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + ((12*A - 8*B + 5*C)*Tan[c + d*x])/(a^2*d) - ((10*A - 7*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((10*A - 7*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((12*A - 8*B + 5*C)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 3768, 3770}"
355,1,237,0,0.5562805,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B+23 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B+23 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((6*A - 13*B + 23*C)*x)/(2*a^3) + (4*(9*A - 19*B + 34*C)*Sin[c + d*x])/(5*a^3*d) - ((6*A - 13*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A - 8*B + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A - 13*B + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) - (4*(9*A - 19*B + 34*C)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,6,41,0.1463,1,"{3041, 2977, 2748, 2635, 8, 2633}"
356,1,207,0,0.5010201,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A-6 B+13 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A-6 B+13 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((2*A - 6*B + 13*C)*x)/(2*a^3) - (2*(11*A - 36*B + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A - 6*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A - 6*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A - 36*B + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,3,41,0.07317,1,"{3041, 2977, 2734}"
357,1,152,0,0.4763898,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(2 A-7 B+27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A+4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(2 A-7 B+27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A+4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((B - 3*C)*x)/a^3 + ((2*A - 7*B + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,7,41,0.1707,1,"{3041, 2977, 2968, 3023, 12, 2735, 2648}"
358,1,123,0,0.2814719,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(6 A+4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(6 A+4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(C*x)/a^3 - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A + 2*B - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((6*A + 4*B - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,5,39,0.1282,1,"{3041, 2968, 3019, 2735, 2648}"
359,1,109,0,0.1322164,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{(2 A+3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(2 A+3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,33,0.09091,1,"{3019, 2750, 2648}"
360,1,124,0,0.3529237,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{(22 A-2 B-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-2 B-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{(22 A-2 B-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-2 B-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 2*B - 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,4,39,0.1026,1,"{3041, 2978, 12, 3770}"
361,1,150,0,0.5167265,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{2 (36 A-11 B+C) \tan (c+d x)}{15 a^3 d}-\frac{(3 A-B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 A-B) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-4 B-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{2 (36 A-11 B+C) \tan (c+d x)}{15 a^3 d}-\frac{(3 A-B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 A-B) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-4 B-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-(((3*A - B)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A - 11*B + C)*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*A - B)*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 8, 3770}"
362,1,210,0,0.5565048,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (76 A-36 B+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A-36 B+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A-6 B+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{2 (76 A-36 B+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A-36 B+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A-6 B+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((13*A - 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A - 36*B + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A - 36*B + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,41,0.1707,1,"{3041, 2978, 2748, 3768, 3770, 3767, 8}"
363,1,246,0,0.5755399,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","\frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-8 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-8 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-((23*A - 13*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) + (4*(34*A - 19*B + 9*C)*Tan[c + d*x])/(5*a^3*d) - ((23*A - 13*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A - 8*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A - 13*B + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(34*A - 19*B + 9*C)*Tan[c + d*x]^3)/(15*a^3*d)","A",8,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 3768, 3770}"
364,1,245,0,0.6910977,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A-8 B+21 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}","-\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A-8 B+21 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"((2*A - 8*B + 21*C)*x)/(2*a^4) - (8*(20*A - 83*B + 216*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((10*A - 52*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(20*A - 83*B + 216*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((B - 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",5,3,41,0.07317,1,"{3041, 2977, 2734}"
365,1,195,0,0.648229,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (B-4 C)}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (B-4 C)}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((B - 4*C)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((2*A + 5*B - 12*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,7,41,0.1707,1,"{3041, 2977, 2968, 3023, 12, 2735, 2648}"
366,1,164,0,0.4783508,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(C*x)/a^4 - ((8*A + 6*B - 55*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((16*A + 12*B - 215*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((4*A + 3*B - 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,6,41,0.1463,1,"{3041, 2977, 2968, 3019, 2735, 2648}"
367,1,148,0,0.3482826,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{(8 A+13 B+36 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-2 B-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(6 A+B-8 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(8 A+13 B+36 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-2 B-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(6 A+B-8 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"((23*A - 2*B - 54*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((8*A + 13*B + 36*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((6*A + B - 8*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",5,5,39,0.1282,1,"{3041, 2968, 3019, 2750, 2648}"
368,1,148,0,0.175611,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",4,4,33,0.1212,1,"{3019, 2750, 2650, 2648}"
369,1,157,0,0.4923434,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^4,x]","-\frac{2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","-\frac{2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 6*B - 8*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (2*(80*A - 3*B - 4*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B - 4*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,4,39,0.1026,1,"{3041, 2978, 12, 3770}"
370,1,185,0,0.7170592,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"-(((4*A - B)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (2*(332*A - 80*B + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((4*A - B)*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B - 2*C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 8, 3770}"
371,1,248,0,0.7612476,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (216 A-83 B+20 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}","-\frac{8 (216 A-83 B+20 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"((21*A - 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B + 20*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A - 52*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(216*A - 83*B + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",9,7,41,0.1707,1,"{3041, 2978, 2748, 3768, 3770, 3767, 8}"
372,1,287,0,0.7982913,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^4,x]","\frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"-((44*A - 21*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x])/(35*a^4*d) - ((44*A - 21*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((178*A - 87*B + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((44*A - 21*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((16*A - 9*B + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x]^3)/(105*a^4*d)","A",9,6,41,0.1463,1,"{3041, 2978, 2748, 3767, 3768, 3770}"
373,1,239,0,0.5469781,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a (99 A+88 B+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+88 B+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+88 B+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}","\frac{2 a (99 A+88 B+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+88 B+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+88 B+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(4*a*(99*A + 88*B + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(11*B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)","A",6,6,43,0.1395,1,"{3045, 2981, 2770, 2759, 2751, 2646}"
374,1,193,0,0.4666577,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (21 A+18 B+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+18 B+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (9 B+C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}","\frac{2 (21 A+18 B+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+18 B+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (9 B+C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(2*a*(21*A + 18*B + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)","A",5,5,43,0.1163,1,"{3045, 2981, 2759, 2751, 2646}"
375,1,147,0,0.347413,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (35 A-14 B+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+49 B+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}","\frac{2 (35 A-14 B+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+49 B+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(2*a*(35*A + 49*B + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 14*B + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*(7*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)","A",5,5,41,0.1220,1,"{3045, 2968, 3023, 2751, 2646}"
376,1,104,0,0.1480875,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}","\frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(2*a*(15*A + 5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",3,3,35,0.08571,1,"{3023, 2751, 2646}"
377,1,100,0,0.274986,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 B+C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 B+C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(2*Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*(3*B + C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,41,0.09756,1,"{3045, 2981, 2773, 206}"
378,1,98,0,0.3056837,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\sqrt{a} (A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}","\frac{\sqrt{a} (A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}",1,"(Sqrt[a]*(A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d","A",4,4,43,0.09302,1,"{3043, 2981, 2773, 206}"
379,1,117,0,0.3479861,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\sqrt{a} (3 A+4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{\sqrt{a} (3 A+4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a]*(3*A + 4*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,43,0.09302,1,"{3043, 2980, 2773, 206}"
380,1,163,0,0.4245526,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (5 A+6 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{a (5 A+6 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(Sqrt[a]*(5*A + 6*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 6*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,43,0.1163,1,"{3043, 2980, 2772, 2773, 206}"
381,1,209,0,0.5085309,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a (35 A+40 B+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+40 B+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+40 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}","\frac{a (35 A+40 B+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+40 B+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+40 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(Sqrt[a]*(35*A + 40*B + 48*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(35*A + 40*B + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,43,0.1163,1,"{3043, 2980, 2772, 2773, 206}"
382,1,243,0,0.7001557,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}","\frac{2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",6,6,43,0.1395,1,"{3045, 2976, 2981, 2759, 2751, 2646}"
383,1,187,0,0.4154208,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+57 B+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}","\frac{8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+57 B+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}",1,"(8*a^2*(63*A + 57*B + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)","A",6,6,41,0.1463,1,"{3045, 2968, 3023, 2751, 2647, 2646}"
384,1,144,0,0.1973764,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{8 a^2 (35 A+21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}","\frac{8 a^2 (35 A+21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}",1,"(8*a^2*(35*A + 21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",4,4,35,0.1143,1,"{3023, 2751, 2647, 2646}"
385,1,142,0,0.4270344,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(15*A + 20*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,41,0.1220,1,"{3045, 2976, 2981, 2773, 206}"
386,1,144,0,0.5000898,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a^2 (3 A-6 B-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (3 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}","-\frac{a^2 (3 A-6 B-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (3 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^(3/2)*(3*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(3*A - 6*B - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",5,5,43,0.1163,1,"{3043, 2976, 2981, 2773, 206}"
387,1,159,0,0.5042235,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (3 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}","-\frac{a^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (3 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a^(3/2)*(7*A + 12*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(5*A + 4*B - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,43,0.1163,1,"{3043, 2975, 2981, 2773, 206}"
388,1,165,0,0.5589277,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (19 A+30 B+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+14 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^2 (19 A+30 B+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+14 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(3/2)*(11*A + 14*B + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(19*A + 30*B + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,43,0.1163,1,"{3043, 2975, 2980, 2773, 206}"
389,1,215,0,0.6418205,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (75 A+88 B+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+88 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (39 A+56 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}","\frac{a^2 (75 A+88 B+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+88 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (39 A+56 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^(3/2)*(75*A + 88*B + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(39*A + 56*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,6,43,0.1395,1,"{3043, 2975, 2980, 2772, 2773, 206}"
390,1,263,0,0.762709,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (133 A+150 B+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^(3/2)*(133*A + 150*B + 176*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(133*A + 150*B + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,6,43,0.1395,1,"{3043, 2975, 2980, 2772, 2773, 206}"
391,1,294,0,0.9593814,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}","\frac{2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(2*a^3*(10439*A + 9230*B + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2522*B + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(13*B + 5*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)","A",7,6,43,0.1395,1,"{3045, 2976, 2981, 2759, 2751, 2646}"
392,1,229,0,0.4937458,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{16 a^2 (165 A+143 B+125 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A-22 B+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (165 A+143 B+125 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 (11 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}","\frac{16 a^2 (165 A+143 B+125 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A-22 B+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (165 A+143 B+125 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 (11 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}",1,"(64*a^3*(165*A + 143*B + 125*C)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(165*A + 143*B + 125*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (2*(11*B + 5*C)*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)","A",7,6,41,0.1463,1,"{3045, 2968, 3023, 2751, 2647, 2646}"
393,1,184,0,0.2451482,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{16 a^2 (21 A+15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}","\frac{16 a^2 (21 A+15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (21 A+15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}",1,"(64*a^3*(21*A + 15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)","A",5,4,35,0.1143,1,"{3023, 2751, 2647, 2646}"
394,1,182,0,0.6615871,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (7 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (7 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(245*A + 224*B + 160*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(35*A + 56*B + 40*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",6,5,41,0.1220,1,"{3045, 2976, 2981, 2773, 206}"
395,1,184,0,0.6761304,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^3 (15 A+70 B+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-10 B-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{a^{5/2} (5 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}","\frac{a^3 (15 A+70 B+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-10 B-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{a^{5/2} (5 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^(5/2)*(5*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*(15*A + 70*B + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 10*B - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d","A",6,5,43,0.1163,1,"{3043, 2976, 2981, 2773, 206}"
396,1,199,0,0.7014363,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{a^3 (27 A-12 B-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A+12 B-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{a^{5/2} (19 A+20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (5 A+4 B) \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}","-\frac{a^3 (27 A-12 B-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A+12 B-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{a^{5/2} (19 A+20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (5 A+4 B) \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}",1,"(a^(5/2)*(19*A + 20*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(27*A - 12*B - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A + 12*B - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (a*(5*A + 4*B)*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,43,0.1395,1,"{3043, 2975, 2976, 2981, 2773, 206}"
397,1,207,0,0.726982,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{a^3 (49 A+54 B-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+42 B+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^{5/2} (25 A+38 B+40 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (5 A+6 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}","-\frac{a^3 (49 A+54 B-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+42 B+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^{5/2} (25 A+38 B+40 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (5 A+6 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}",1,"(a^(5/2)*(25*A + 38*B + 40*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(49*A + 54*B - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 42*B + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(5*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,5,43,0.1163,1,"{3043, 2975, 2981, 2773, 206}"
398,1,215,0,0.7895238,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^3 (299 A+392 B+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+200 B+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (17 A+24 B+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (5 A+8 B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}","\frac{a^3 (299 A+392 B+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+200 B+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (17 A+24 B+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (5 A+8 B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^(5/2)*(163*A + 200*B + 304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(299*A + 392*B + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 24*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (a*(5*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,43,0.1163,1,"{3043, 2975, 2980, 2773, 206}"
399,1,261,0,0.9024845,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^3 (283 A+326 B+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+326 B+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (79 A+110 B+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^3 (787 A+950 B+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+2 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}","\frac{a^3 (283 A+326 B+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+326 B+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (79 A+110 B+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^3 (787 A+950 B+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+2 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^(5/2)*(283*A + 326*B + 400*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*(A + 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,6,43,0.1395,1,"{3043, 2975, 2980, 2772, 2773, 206}"
400,1,311,0,0.9664082,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^2 (115 A+156 B+120 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \tan (c+d x) \sec ^2(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a (5 A+12 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}","\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^2 (115 A+156 B+120 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^3 (545 A+628 B+680 C) \tan (c+d x) \sec ^2(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a (5 A+12 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1015*A + 1132*B + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Sec[c + d*x]^2*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(480*d) + (a*(5*A + 12*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(60*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",8,6,43,0.1395,1,"{3043, 2975, 2980, 2772, 2773, 206}"
401,1,254,0,0.8379265,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (21 A-3 B+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A-93 B+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (9 B-C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (21 A-3 B+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A-93 B+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (9 B-C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A - 111*B + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(9*B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)","A",8,7,43,0.1628,1,"{3045, 2983, 2968, 3023, 2751, 2649, 206}"
402,1,208,0,0.6117909,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (35 A-7 B+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (7 B-C) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (35 A-7 B+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (7 B-C) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A - 49*B + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 7*B + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)","A",7,7,43,0.1628,1,"{3045, 2983, 2968, 3023, 2751, 2649, 206}"
403,1,164,0,0.3802342,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (15 A-10 B+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (5 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (15 A-10 B+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (5 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A - 10*B + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)","A",6,6,41,0.1463,1,"{3045, 2968, 3023, 2751, 2649, 206}"
404,1,118,0,0.1582691,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",4,4,35,0.1143,1,"{3023, 2751, 2649, 206}"
405,1,118,0,0.314117,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,41,0.1220,1,"{3045, 2985, 2649, 206, 2773}"
406,1,120,0,0.3512394,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"-(((A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,43,0.1163,1,"{3043, 2985, 2649, 206, 2773}"
407,1,169,0,0.540153,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(7 A-4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{(7 A-4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((7*A - 4*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,43,0.1395,1,"{3043, 2984, 2985, 2649, 206, 2773}"
408,1,213,0,0.7369827,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(7 A-2 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A-14 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{(7 A-2 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A-14 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"-((9*A - 14*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,43,0.1395,1,"{3043, 2984, 2985, 2649, 206, 2773}"
409,1,259,0,0.9341816,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(21 A-56 B+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}","-\frac{(21 A-56 B+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}",1,"((107*A - 72*B + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A - 8*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,43,0.1395,1,"{3043, 2984, 2985, 2649, 206, 2773}"
410,1,277,0,0.8725872,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(245 A-273 B+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A-15 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}","\frac{(245 A-273 B+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A-15 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 15*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A - 651*B + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A - 63*B + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A - 7*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A - 273*B + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)","A",8,7,43,0.1628,1,"{3041, 2983, 2968, 3023, 2751, 2649, 206}"
411,1,229,0,0.6695422,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(15 A-35 B+39 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A-35 B+39 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}",1,"-((7*A - 11*B + 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((45*A - 65*B + 93*C)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A - 5*B + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((15*A - 35*B + 39*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)","A",7,7,43,0.1628,1,"{3041, 2983, 2968, 3023, 2751, 2649, 206}"
412,1,181,0,0.3960363,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 A-3 B+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A-7 B+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A-9 B+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}","\frac{(3 A-3 B+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(3 A-7 B+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A-9 B+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"((3*A - 7*B + 11*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A - 9*B + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A - 3*B + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)","A",6,6,41,0.1463,1,"{3041, 2968, 3023, 2751, 2649, 206}"
413,1,120,0,0.1618378,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A+3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}","\frac{(A+3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((A + 3*B - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{3019, 2751, 2649, 206}"
414,1,131,0,0.3480428,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(5 A-B-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(5 A-B-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - B - 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,41,0.1220,1,"{3041, 2985, 2649, 206, 2773}"
415,1,173,0,0.5671646,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(9 A-5 B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(3 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A-B+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(9 A-5 B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(3 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A-B+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-(((3*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,43,0.1395,1,"{3041, 2984, 2985, 2649, 206, 2773}"
416,1,232,0,0.7799116,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A-12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A-9 B+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A-6 B+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(19 A-12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A-9 B+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A-6 B+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A - 12*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B + 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((7*A - 6*B + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,43,0.1395,1,"{3041, 2984, 2985, 2649, 206, 2773}"
417,1,284,0,0.9921749,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(47 A-38 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A-13 B+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A-3 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A-12 B+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(47 A-38 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A-13 B+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A-3 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A-12 B+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}",1,"-((47*A - 38*B + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*a^(3/2)*d) + ((17*A - 13*B + 9*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((21*A - 14*B + 12*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A - 12*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,43,0.1395,1,"{3041, 2984, 2985, 2649, 206, 2773}"
418,1,277,0,0.8977226,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(45 A-85 B+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(195 A-475 B+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(465 A-985 B+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-163 B+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(45 A-85 B+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(195 A-475 B+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(465 A-985 B+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-163 B+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"-((75*A - 163*B + 283*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A - 85*B + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)","A",8,8,43,0.1860,1,"{3041, 2977, 2983, 2968, 3023, 2751, 2649, 206}"
419,1,227,0,0.6868076,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(15 A-39 B+95 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(15 A-39 B+95 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A - 75*B + 163*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A - 9*B + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A - 93*B + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((15*A - 39*B + 95*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)","A",7,7,43,0.1628,1,"{3041, 2977, 2968, 3023, 2751, 2649, 206}"
420,1,179,0,0.4074424,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(A-B+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A+19 B-75 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(A-B+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A+19 B-75 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((5*A + 19*B - 75*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((3*A + 5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - B + 9*C)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,41,0.1463,1,"{3041, 2968, 3019, 2751, 2649, 206}"
421,1,133,0,0.1768498,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A+5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(3 A+5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((3*A + 5*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,35,0.1143,1,"{3019, 2750, 2649, 206}"
422,1,171,0,0.5209448,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(43 A-3 B-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-3 B-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{(43 A-3 B-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-3 B-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B - 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,41,0.1463,1,"{3041, 2978, 2985, 2649, 206, 2773}"
423,1,217,0,0.8022304,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(35 A-11 B+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(115 A-43 B+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(5 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(15 A-7 B-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(35 A-11 B+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(115 A-43 B+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(5 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(15 A-7 B-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-(((5*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B + 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,43,0.1628,1,"{3041, 2978, 2984, 2985, 2649, 206, 2773}"
424,1,280,0,1.0181123,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(63 A-35 B+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A-20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A-115 B+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(31 A-15 B+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{(63 A-35 B+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A-20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A-115 B+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(31 A-15 B+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((39*A - 20*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B + 43*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((63*A - 35*B + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,43,0.1628,1,"{3041, 2978, 2984, 2985, 2649, 206, 2773}"
425,1,123,0,0.1166293,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,5,31,0.1613,1,"{3023, 2748, 2635, 2641, 2639}"
426,1,93,0,0.1005843,"\int \sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,31,0.1613,1,"{3023, 2748, 2639, 2635, 2641}"
427,1,65,0,0.0865771,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,31,0.1290,1,"{3023, 2748, 2641, 2639}"
428,1,61,0,0.0881948,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,31,0.1290,1,"{3021, 2748, 2641, 2639}"
429,1,87,0,0.1016879,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","\frac{2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,31,0.1613,1,"{3021, 2748, 2636, 2639, 2641}"
430,1,123,0,0.1197698,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2),x]","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,5,31,0.1613,1,"{3021, 2748, 2636, 2641, 2639}"
431,1,211,0,0.2872962,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 a (11 A+11 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+11 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+11 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}","\frac{10 a (11 A+11 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+11 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+11 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}",1,"(2*a*(9*A + 7*(B + C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(11*A + 11*B + 9*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*a*(11*A + 11*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*(B + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 11*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*(B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)","A",8,6,41,0.1463,1,"{3033, 3023, 2748, 2635, 2639, 2641}"
432,1,177,0,0.2674005,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a (7 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+9 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{2 a (7 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+9 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*a*(9*A + 9*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*A + 5*(B + C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 9*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",7,6,41,0.1463,1,"{3033, 3023, 2748, 2635, 2641, 2639}"
433,1,144,0,0.2284589,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 a (7 A+7 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+7 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 a (7 A+7 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+7 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(2*a*(5*A + 3*(B + C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 7*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,6,41,0.1463,1,"{3033, 3023, 2748, 2639, 2635, 2641}"
434,1,107,0,0.1968721,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 a (3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 a (3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*a*(5*A + 5*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,41,0.1220,1,"{3033, 3023, 2748, 2641, 2639}"
435,1,101,0,0.2013028,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 a (3 A+3 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a (3 A+3 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*a*(A - B - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + 3*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,41,0.1220,1,"{3031, 3023, 2748, 2641, 2639}"
436,1,100,0,0.2189421,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a (A+3 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a (A+3 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*a*(A + B - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*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",5,5,41,0.1220,1,"{3031, 3021, 2748, 2641, 2639}"
437,1,139,0,0.2542232,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a (A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 (B+C)) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 (B+C)) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*a*(3*A + 5*(B + C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*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*(3*A + 5*(B + C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,6,41,0.1463,1,"{3031, 3021, 2748, 2636, 2639, 2641}"
438,1,177,0,0.2706463,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 (B+C)) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a (5 A+7 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 (B+C)) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*a*(3*A + 3*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*(B + C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*(B + C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 3*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,41,0.1463,1,"{3031, 3021, 2748, 2636, 2641, 2639}"
439,1,251,0,0.5372055,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{4 a^2 (66 A+55 B+50 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+4 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}","\frac{4 a^2 (66 A+55 B+50 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+4 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"(4*a^2*(9*A + 8*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 8*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 121*B + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(11*B + 4*C)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)","A",9,8,43,0.1860,1,"{3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
440,1,215,0,0.5044268,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{4 a^2 (7 A+6 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 (9 B+4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}","\frac{4 a^2 (7 A+6 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 (9 B+4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(4*a^2*(12*A + 9*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*B + 4*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)","A",8,8,43,0.1860,1,"{3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
441,1,179,0,0.477012,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{4 a^2 (14 A+7 B+6 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 B+4 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}","\frac{4 a^2 (14 A+7 B+6 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 B+4 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"(4*a^2*(5*A + 4*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*B + 4*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",7,7,43,0.1628,1,"{3045, 2976, 2968, 3023, 2748, 2641, 2639}"
442,1,172,0,0.479432,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{4 a^2 (3 A+2 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \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 (5 B+4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}","\frac{4 a^2 (3 A+2 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \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 (5 B+4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"(4*a^2*(5*B + 4*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",7,7,43,0.1628,1,"{3043, 2976, 2968, 3023, 2748, 2641, 2639}"
443,1,172,0,0.4715992,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{4 a^2 (2 A+3 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{4 a^2 (2 A+3 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(4*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",7,7,43,0.1628,1,"{3043, 2975, 2968, 3023, 2748, 2641, 2639}"
444,1,174,0,0.4870779,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{4 a^2 (A+2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^2 (A+2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(4*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",7,7,43,0.1628,1,"{3043, 2975, 2968, 3021, 2748, 2641, 2639}"
445,1,215,0,0.5240241,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{4 a^2 (6 A+7 B+14 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{4 a^2 (6 A+7 B+14 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^2*(3*A + 4*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))","A",8,8,43,0.1860,1,"{3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
446,1,251,0,0.5518779,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{4 a^2 (5 A+6 B+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+9 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{4 a^2 (5 A+6 B+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+9 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-4*a^2*(8*A + 9*B + 12*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 6*B + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(8*A + 9*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(4*A + 9*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))","A",9,8,43,0.1860,1,"{3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
447,1,303,0,0.7347936,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{4 a^3 (121 A+105 B+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (13 B+6 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}","\frac{4 a^3 (121 A+105 B+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (13 B+6 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}",1,"(4*a^3*(221*A + 195*B + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 195*B + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (20*a^3*(286*A + 273*B + 236*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (2*(13*B + 6*C)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(143*A + 195*B + 145*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)","A",10,8,43,0.1860,1,"{3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
448,1,267,0,0.6850366,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{4 a^3 (143 A+121 B+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+6 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}","\frac{4 a^3 (143 A+121 B+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+6 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(4*a^3*(21*A + 17*B + 15*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(11*B + 6*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(99*A + 143*B + 105*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d)","A",9,8,43,0.1860,1,"{3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
449,1,231,0,0.6596886,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{4 a^3 (21 A+13 B+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}","\frac{4 a^3 (21 A+13 B+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"(4*a^3*(27*A + 21*B + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)","A",8,7,43,0.1628,1,"{3045, 2976, 2968, 3023, 2748, 2641, 2639}"
450,1,229,0,0.6723392,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{4 a^3 (35 A+21 B+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}","\frac{4 a^3 (35 A+21 B+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"(4*a^3*(5*A + 9*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(35*A - 7*B - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",8,7,43,0.1628,1,"{3043, 2976, 2968, 3023, 2748, 2641, 2639}"
451,1,227,0,0.6610954,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{4 a^3 (5 A+5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{2 (2 A+B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{4 a^3 (5 A+5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{2 (2 A+B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^3*(5*A - 5*B - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(35*A + 15*B - 3*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",8,8,43,0.1860,1,"{3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
452,1,230,0,0.6779167,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{4 a^3 (3 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^3 (3 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a^3*(9*A + 5*B - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(33*A + 35*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]])","A",8,7,43,0.1628,1,"{3043, 2975, 2968, 3023, 2748, 2641, 2639}"
453,1,231,0,0.6689714,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{4 a^3 (13 A+21 B+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{4 a^3 (13 A+21 B+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^3*(7*A + 9*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(6*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",8,7,43,0.1628,1,"{3043, 2975, 2968, 3021, 2748, 2641, 2639}"
454,1,267,0,0.7022396,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{4 a^3 (11 A+13 B+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{4 a^3 (11 A+13 B+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-4*a^3*(17*A + 21*B + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))","A",9,8,43,0.1860,1,"{3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
455,1,303,0,0.7487622,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{4 a^3 (105 A+121 B+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+11 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{4 a^3 (105 A+121 B+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+11 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(-4*a^3*(15*A + 17*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 121*B + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(15*A + 17*B + 21*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (2*(6*A + 11*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d*Cos[c + d*x]^(9/2)) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2))","A",10,8,43,0.1860,1,"{3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
456,1,210,0,0.2525363,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{5 (7 A-7 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-7 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A-7 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}","\frac{5 (7 A-7 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-7 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A-7 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(-3*(5*A - 7*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A - 7*B + 9*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A - 7*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A - 7*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,5,43,0.1163,1,"{3041, 2748, 2635, 2639, 2641}"
457,1,174,0,0.2336073,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{(3 A-5 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-5 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A-5 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","-\frac{(3 A-5 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-5 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A-5 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(3*(5*A - 5*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A - 5*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,5,43,0.1163,1,"{3041, 2748, 2635, 2641, 2639}"
458,1,134,0,0.2033297,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A-3 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","\frac{(3 A-3 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"-(((A - 3*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((3*A - 3*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,43,0.1163,1,"{3041, 2748, 2639, 2635, 2641}"
459,1,90,0,0.1831672,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{(A+B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{(A+B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((A - B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A + B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,43,0.09302,1,"{3041, 2748, 2641, 2639}"
460,1,125,0,0.2071552,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{(A-B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}","-\frac{(A-B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"-(((3*A - B + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A - B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",5,5,43,0.1163,1,"{3041, 2748, 2636, 2639, 2641}"
461,1,165,0,0.2287161,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{(5 A-3 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-3 B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","\frac{(5 A-3 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-3 B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"((3*A - 3*B + C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A - 3*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A - 3*B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",6,5,43,0.1163,1,"{3041, 2748, 2636, 2641, 2639}"
462,1,210,0,0.2469298,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]","-\frac{(5 A-5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A-5 B+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}","-\frac{(5 A-5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A-5 B+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(-3*(7*A - 5*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A - 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A - 5*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))","A",7,5,43,0.1163,1,"{3041, 2748, 2636, 2639, 2641}"
463,1,214,0,0.40253,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{5 (A-2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-2 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(20 A-35 B+56 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A-2 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 (A-2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-2 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(20 A-35 B+56 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A-2 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((20*A - 35*B + 56*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((20*A - 35*B + 56*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A - 2*B + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,43,0.1395,1,"{3041, 2977, 2748, 2635, 2641, 2639}"
464,1,180,0,0.375277,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{(2 A-5 B+10 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-4 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-5 B+10 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 A-5 B+10 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-4 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-5 B+10 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((A - 4*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A - 5*B + 10*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A - 4*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,43,0.1395,1,"{3041, 2977, 2748, 2639, 2635, 2641}"
465,1,139,0,0.34334,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{(A+2 B-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+2 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A+2 B-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+2 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((B - 4*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((A + 2*B - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,43,0.1163,1,"{3041, 2977, 2748, 2641, 2639}"
466,1,133,0,0.3473993,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{(2 A+B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 A+B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A + B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,43,0.1163,1,"{3041, 2978, 2748, 2641, 2639}"
467,1,175,0,0.3763581,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{(5 A-2 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-2 B-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(5 A-2 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-2 B-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"-(((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) - ((5*A - 2*B - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - 2*B - C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",6,6,43,0.1395,1,"{3041, 2978, 2748, 2636, 2639, 2641}"
468,1,211,0,0.4021628,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{(10 A-5 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(10 A-5 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((7*A - 4*B + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((10*A - 5*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",7,6,43,0.1395,1,"{3041, 2978, 2748, 2636, 2641, 2639}"
469,1,273,0,0.6022285,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{(13 A-33 B+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A-17 B+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{(13 A-33 B+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A-17 B+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(7*(7*A - 17*B + 33*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 33*B + 63*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A - 17*B + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - 7*B + 12*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 33*B + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{3041, 2977, 2748, 2635, 2641, 2639}"
470,1,232,0,0.5747879,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(3 A-13 B+33 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-49 B+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","\frac{(3 A-13 B+33 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-49 B+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"-((9*A - 49*B + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B + 33*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((B - 2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - 49*B + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{3041, 2977, 2748, 2639, 2635, 2641}"
471,1,195,0,0.5302054,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+3 B-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+3 B-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(2 A+3 B-8 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(A+3 B-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+3 B-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(2 A+3 B-8 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((A + 9*B - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,43,0.1163,1,"{3041, 2977, 2748, 2641, 2639}"
472,1,191,0,0.5242322,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}","\frac{(A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"((A - B - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{3041, 2977, 2978, 2748, 2641, 2639}"
473,1,193,0,0.5304487,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{(3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((9*A + B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,43,0.1163,1,"{3041, 2978, 2748, 2641, 2639}"
474,1,237,0,0.5803557,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{(13 A-3 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(8 A-3 B-2 C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{(13 A-3 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(8 A-3 B-2 C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"-((49*A - 9*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - 9*B - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{3041, 2978, 2748, 2636, 2639, 2641}"
475,1,270,0,0.6149395,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{(33 A-13 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(33 A-13 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((119*A - 49*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{3041, 2978, 2748, 2636, 2641, 2639}"
476,1,227,0,0.5281398,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (48 A+40 B+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}","\frac{a (48 A+40 B+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(Sqrt[a]*(48*A + 40*B + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(48*A + 40*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,45,0.1111,1,"{3045, 2981, 2770, 2774, 216}"
477,1,179,0,0.4306035,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{a} (8 A+6 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{\sqrt{a} (8 A+6 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(Sqrt[a]*(8*A + 6*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(8*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{3045, 2981, 2770, 2774, 216}"
478,1,131,0,0.3552257,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a} (8 A+4 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{\sqrt{a} (8 A+4 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a]*(8*A + 4*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(4*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,45,0.08889,1,"{3045, 2981, 2774, 216}"
479,1,121,0,0.3574506,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} (2 B+C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} (2 B+C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*(2*B + C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,45,0.08889,1,"{3043, 2981, 2774, 216}"
480,1,120,0,0.3425263,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,45,0.08889,1,"{3043, 2980, 2774, 216}"
481,1,130,0,0.368801,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a (8 A+10 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (8 A+10 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*a*(A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 10*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",3,3,45,0.06667,1,"{3043, 2980, 2771}"
482,1,178,0,0.4392643,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a*(A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,45,0.08889,1,"{3043, 2980, 2772, 2771}"
483,1,226,0,0.5181748,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a*(A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,45,0.08889,1,"{3043, 2980, 2772, 2771}"
484,1,283,0,0.7516865,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^(3/2)*(176*A + 150*B + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(176*A + 150*B + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 90*B + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(10*B + 3*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,45,0.1333,1,"{3045, 2976, 2981, 2770, 2774, 216}"
485,1,233,0,0.6448503,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}","\frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^(3/2)*(112*A + 88*B + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(112*A + 88*B + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(48*A + 56*B + 39*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + 3*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,45,0.1333,1,"{3045, 2976, 2981, 2770, 2774, 216}"
486,1,181,0,0.5655717,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^{3/2} (24 A+14 B+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^{3/2} (24 A+14 B+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(3/2)*(24*A + 14*B + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(2*B + C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{3045, 2976, 2981, 2774, 216}"
487,1,181,0,0.5802645,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^{3/2} (8 A+12 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (8 A+12 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^(3/2)*(8*A + 12*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,45,0.1111,1,"{3043, 2976, 2981, 2774, 216}"
488,1,171,0,0.576854,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (2 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (2 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^(3/2)*(2*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(8*A + 6*B - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",5,5,45,0.1111,1,"{3043, 2975, 2981, 2774, 216}"
489,1,172,0,0.5301716,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",5,5,45,0.1111,1,"{3043, 2975, 2980, 2774, 216}"
490,1,184,0,0.5792028,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a^2*(4*A + 6*B + 5*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(104*A + 126*B + 175*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,45,0.08889,1,"{3043, 2975, 2980, 2771}"
491,1,232,0,0.6790721,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^2*(52*A + 72*B + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,5,45,0.1111,1,"{3043, 2975, 2980, 2772, 2771}"
492,1,284,0,0.7633655,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",6,5,45,0.1111,1,"{3043, 2975, 2980, 2772, 2771}"
493,1,333,0,0.9899662,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}","\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(680*A + 628*B + 545*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,45,0.1333,1,"{3045, 2976, 2981, 2770, 2774, 216}"
494,1,281,0,0.8814708,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^{5/2} (400 A+326 B+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}","\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a^{5/2} (400 A+326 B+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^(5/2)*(400*A + 326*B + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(400*A + 326*B + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 950*B + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(2*B + C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,45,0.1333,1,"{3045, 2976, 2981, 2770, 2774, 216}"
495,1,233,0,0.7853496,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}","\frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^(5/2)*(304*A + 200*B + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(8*B + 5*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,45,0.1111,1,"{3045, 2976, 2981, 2774, 216}"
496,1,231,0,0.7931083,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^{5/2} (40 A+38 B+25 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (40 A+38 B+25 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^(5/2)*(40*A + 38*B + 25*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,5,45,0.1111,1,"{3043, 2976, 2981, 2774, 216}"
497,1,233,0,0.7988765,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a^{5/2} (8 A+20 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a (5 A+3 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (8 A+20 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a (5 A+3 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^(5/2)*(8*A + 20*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,45,0.1333,1,"{3043, 2975, 2976, 2981, 2774, 216}"
498,1,223,0,0.8088355,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{a^{5/2} (2 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{a^{5/2} (2 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^(5/2)*(2*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(64*A + 70*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,5,45,0.1111,1,"{3043, 2975, 2981, 2774, 216}"
499,1,222,0,0.7244329,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 A+7 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 A+7 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",6,5,45,0.1111,1,"{3043, 2975, 2980, 2774, 216}"
500,1,234,0,0.8087793,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+9 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+9 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^3*(8*A + 10*B + 11*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(584*A + 690*B + 903*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 90*B + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 9*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,45,0.08889,1,"{3043, 2975, 2980, 2771}"
501,1,284,0,0.8952956,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(2*a^3*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",6,5,45,0.1111,1,"{3043, 2975, 2980, 2772, 2771}"
502,1,334,0,0.9893486,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}",1,"(2*a^3*(2224*A + 2522*B + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))","A",7,5,45,0.1111,1,"{3043, 2975, 2980, 2772, 2771}"
503,1,241,0,0.8451507,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(8 A-14 B+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","-\frac{(8 A-14 B+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"-((8*A - 14*B + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A - 2*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + ((6*B - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,45,0.1556,1,"{3045, 2983, 2982, 2782, 205, 2774, 216}"
504,1,195,0,0.6250716,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(8 A-4 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{(8 A-4 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((8*A - 4*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{3045, 2983, 2982, 2782, 205, 2774, 216}"
505,1,141,0,0.439583,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"((2*B - C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{3045, 2982, 2782, 205, 2774, 216}"
506,1,138,0,0.4245355,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{3043, 2982, 2782, 205, 2774, 216}"
507,1,143,0,0.3773528,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,45,0.1111,1,"{3043, 2984, 12, 2782, 205}"
508,1,191,0,0.5592691,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (13 A-5 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 (13 A-5 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A - 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,45,0.1111,1,"{3043, 2984, 12, 2782, 205}"
509,1,237,0,0.7599883,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A - 91*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,45,0.1111,1,"{3043, 2984, 12, 2782, 205}"
510,1,213,0,0.7532146,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(8 a A-4 a B-4 A b+7 b B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{(4 a B+4 A b-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (a-b) (A-B) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{(8 a A-4 a B-4 A b+7 b B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{(4 a B+4 A b-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (a-b) (A-B) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((8*a*A - 4*A*b - 4*a*B + 7*b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*A*b + 4*a*B - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,54,0.1296,1,"{3045, 2983, 2982, 2782, 205, 2774, 216}"
511,1,260,0,0.86877,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(8 A-12 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A-6 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}","\frac{(8 A-12 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A-6 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}",1,"((8*A - 12*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A - 6*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,45,0.1556,1,"{3041, 2983, 2982, 2782, 205, 2774, 216}"
512,1,202,0,0.6267334,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A-5 B+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}","\frac{(A-5 B+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"((2*B - 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{3041, 2983, 2982, 2782, 205, 2774, 216}"
513,1,149,0,0.4336562,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(3 A+B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A+B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,6,45,0.1333,1,"{3041, 2982, 2782, 205, 2774, 216}"
514,1,161,0,0.4180862,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(7 A-3 B-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-3 B-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,45,0.1111,1,"{3041, 2984, 12, 2782, 205}"
515,1,213,0,0.598038,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(11 A-7 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(11 A-7 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,45,0.1111,1,"{3041, 2984, 12, 2782, 205}"
516,1,263,0,0.8207978,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(15 A-11 B+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(39 A-35 B+15 C) \sin (c+d x)}{30 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A-5 B+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(147 A-95 B+75 C) \sin (c+d x)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A-11 B+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(39 A-35 B+15 C) \sin (c+d x)}{30 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A-5 B+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(147 A-95 B+75 C) \sin (c+d x)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"-((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sin[c + d*x])/(30*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((147*A - 95*B + 75*C)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,45,0.1111,1,"{3041, 2984, 12, 2782, 205}"
517,1,254,0,0.8718903,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A-11 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(3 A-43 B+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A+7 B-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A-11 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(3 A-43 B+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A+7 B-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((2*B - 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,8,45,0.1778,1,"{3041, 2977, 2983, 2982, 2782, 205, 2774, 216}"
518,1,201,0,0.630439,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(5 A+3 B-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(5 A+3 B-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,7,45,0.1556,1,"{3041, 2977, 2982, 2782, 205, 2774, 216}"
519,1,163,0,0.4413383,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(19 A+5 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+5 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,45,0.1111,1,"{3041, 2978, 12, 2782, 205}"
520,1,211,0,0.6397837,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(49 A-9 B+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A-9 B+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"-((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{3041, 2978, 2984, 12, 2782, 205}"
521,1,261,0,0.849588,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(95 A-39 B+15 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-75 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A-9 B+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(95 A-39 B+15 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-75 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A-9 B+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A - 147*B + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,6,45,0.1333,1,"{3041, 2978, 2984, 12, 2782, 205}"
522,1,131,0,0.1953737,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}","\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(4*A + 3*C)*x)/8 + (b*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (b*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{3034, 3023, 2748, 2635, 8, 2633}"
523,1,108,0,0.1078716,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x (4 A+3 C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x (4 A+3 C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(b*(4*A + 3*C)*x)/8 + (a*(3*A + 2*C)*Sin[c + d*x])/(3*d) + (b*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",3,3,29,0.1034,1,"{3034, 3023, 2734}"
524,1,96,0,0.0736837,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","-\frac{\left(a^2 C-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}","-\frac{\left(a^2 C-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}",1,"(a*(2*A + C)*x)/2 - ((a^2*C - b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)","A",2,2,23,0.08696,1,"{3024, 2734}"
525,1,58,0,0.1158787,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+\frac{1}{2} b x (2 A+C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+\frac{1}{2} b x (2 A+C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(b*(2*A + C)*x)/2 + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,29,0.1379,1,"{3034, 3023, 2735, 3770}"
526,1,42,0,0.106029,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+a C x+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}","\frac{a A \tan (c+d x)}{d}+a C x+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}",1,"a*C*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d","A",4,4,31,0.1290,1,"{3032, 3023, 2735, 3770}"
527,1,58,0,0.1325298,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{A b \tan (c+d x)}{d}+b C x","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{A b \tan (c+d x)}{d}+b C x",1,"b*C*x + (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*b*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,31,0.1290,1,"{3032, 3021, 2735, 3770}"
528,1,86,0,0.1728457,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(b*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (A*b*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{3032, 3021, 2748, 3767, 8, 3770}"
529,1,117,0,0.1908691,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{3032, 3021, 2748, 3768, 3770, 3767, 8}"
530,1,140,0,0.2035931,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a (4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (3 A+4 C) \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 (4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (3 A+4 C) \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,"(b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(4*A + 5*C)*Tan[c + d*x])/(5*d) + (b*(3*A + 4*C)*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*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(4*A + 5*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{3032, 3021, 2748, 3767, 3768, 3770}"
531,1,214,0,0.4893745,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{\left(2 a^2 C+b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)-\frac{2 a b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a b C \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}","\frac{\left(2 a^2 C+b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)-\frac{2 a b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a b C \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}",1,"((b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*x)/16 + (2*a*b*(5*A + 4*C)*Sin[c + d*x])/(5*d) + ((b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((2*a^2*C + b^2*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*b*C*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - (2*a*b*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)","A",8,7,33,0.2121,1,"{3050, 3033, 3023, 2748, 2635, 8, 2633}"
532,1,178,0,0.2964653,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{\left(5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{\left(2 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{a b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x (4 A+3 C)+\frac{a b C \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\left(5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{\left(2 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{a b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x (4 A+3 C)+\frac{a b C \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(a*b*(4*A + 3*C)*x)/4 + ((5*a^2*(3*A + 2*C) + 2*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + (a*b*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((2*a^2*C + b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*b*C*Cos[c + d*x]^3*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d)","A",4,4,31,0.1290,1,"{3050, 3033, 3023, 2734}"
533,1,161,0,0.2145356,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a \left(a^2 (-C)+12 A b^2+8 b^2 C\right) \sin (c+d x)}{6 b d}-\frac{\left(2 a^2 C-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}","\frac{a \left(a^2 (-C)+12 A b^2+8 b^2 C\right) \sin (c+d x)}{6 b d}-\frac{\left(2 a^2 C-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}",1,"((4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x)/8 + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Sin[c + d*x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)","A",3,3,25,0.1200,1,"{3024, 2753, 2734}"
534,1,103,0,0.2785641,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\left(2 C \left(a^2+b^2\right)+3 A b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac{a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{\left(2 C \left(a^2+b^2\right)+3 A b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac{a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"a*b*(2*A + C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 2*(a^2 + b^2)*C)*Sin[c + d*x])/(3*d) + (a*b*C*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{3050, 3033, 3023, 2735, 3770}"
535,1,109,0,0.3147269,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{1}{2} x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{2 a b (A-C) \sin (c+d x)}{d}+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}","\frac{1}{2} x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{2 a b (A-C) \sin (c+d x)}{d}+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*A*b^2 + (2*a^2 + b^2)*C)*x)/2 + (2*a*A*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(A - C)*Sin[c + d*x])/d - (b^2*(2*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d","A",5,5,33,0.1515,1,"{3048, 3033, 3023, 2735, 3770}"
536,1,103,0,0.3127126,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A b \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+2 a b C x-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}","\frac{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A b \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+2 a b C x-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}",1,"2*a*b*C*x + ((2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(A - 2*C)*Sin[c + d*x])/(2*d) + (a*A*b*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,33,0.1515,1,"{3048, 3031, 3023, 2735, 3770}"
537,1,112,0,0.3145078,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(a^2 (2 A+3 C)+2 A b^2\right) \tan (c+d x)}{3 d}+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x","\frac{\left(a^2 (2 A+3 C)+2 A b^2\right) \tan (c+d x)}{3 d}+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x",1,"b^2*C*x + (a*b*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (a*A*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,33,0.1515,1,"{3048, 3031, 3021, 2735, 3770}"
538,1,154,0,0.4216702,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{\left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"((4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*(2*A + 3*C)*Tan[c + d*x])/(3*d) + ((2*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,33,0.2121,1,"{3048, 3031, 3021, 2748, 3767, 8, 3770}"
539,1,187,0,0.4467143,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{\left(a^2 (4 A+5 C)+2 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{a A b \tan (c+d x) \sec ^3(c+d x)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{\left(a^2 (4 A+5 C)+2 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{a A b \tan (c+d x) \sec ^3(c+d x)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(a*b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(4*d) + ((5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (a*b*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + ((2*A*b^2 + a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*A*b*Sec[c + d*x]^3*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,33,0.2424,1,"{3048, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
540,1,264,0,0.5393267,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{a \left(5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{b \left(6 a^2 C+5 b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac{a \left(C \left(a^2+12 b^2\right)+15 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right)+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}","\frac{a \left(5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{b \left(6 a^2 C+5 b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac{a \left(C \left(a^2+12 b^2\right)+15 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right)+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}",1,"(b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*x)/16 + (a*(5*a^2*(3*A + 2*C) + 6*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + (b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(15*A*b^2 + (a^2 + 12*b^2)*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(6*a^2*C + 5*b^2*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d)","A",5,5,31,0.1613,1,"{3050, 3049, 3033, 3023, 2734}"
541,1,225,0,0.3409411,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","-\frac{\left(-4 a^2 b^2 (20 A+13 C)+3 a^4 C-4 b^4 (5 A+4 C)\right) \sin (c+d x)}{30 b d}-\frac{\left(3 a^2 C-4 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{a \left(-6 a^2 C+100 A b^2+71 b^2 C\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}","-\frac{\left(-4 a^2 b^2 (20 A+13 C)+3 a^4 C-4 b^4 (5 A+4 C)\right) \sin (c+d x)}{30 b d}-\frac{\left(3 a^2 C-4 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{a \left(-6 a^2 C+100 A b^2+71 b^2 C\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}",1,"(a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*x)/8 - ((3*a^4*C - 4*b^4*(5*A + 4*C) - 4*a^2*b^2*(20*A + 13*C))*Sin[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) - ((3*a^2*C - 4*b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)","A",4,3,25,0.1200,1,"{3024, 2753, 2734}"
542,1,167,0,0.5424269,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a \left(C \left(a^2+4 b^2\right)+6 A b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(12 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{a \left(C \left(a^2+4 b^2\right)+6 A b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(12 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(b*(12*a^2*(2*A + C) + b^2*(4*A + 3*C))*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (a*(6*A*b^2 + (a^2 + 4*b^2)*C)*Sin[c + d*x])/(2*d) + (b*(2*a^2*C + b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",6,6,31,0.1935,1,"{3050, 3049, 3033, 3023, 2735, 3770}"
543,1,167,0,0.5036155,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{b \left(a^2 (6 A-8 C)-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2 C+6 A b^2+3 b^2 C\right)+\frac{3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^2 (6 A-5 C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}","-\frac{b \left(a^2 (6 A-8 C)-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2 C+6 A b^2+3 b^2 C\right)+\frac{3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^2 (6 A-5 C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"(a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*x)/2 + (3*a^2*A*b*ArcTanh[Sin[c + d*x]])/d - (b*(a^2*(6*A - 8*C) - b^2*(3*A + 2*C))*Sin[c + d*x])/(3*d) - (a*b^2*(6*A - 5*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d","A",6,6,33,0.1818,1,"{3048, 3049, 3033, 3023, 2735, 3770}"
544,1,168,0,0.5796876,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a \left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{3 A b \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a \left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{3 A b \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(b*(2*A*b^2 + (6*a^2 + b^2)*C)*x)/2 + (a*(6*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a*b^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) - (b^3*(4*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*A*b*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,33,0.1818,1,"{3048, 3047, 3033, 3023, 2735, 3770}"
545,1,163,0,0.5359143,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+3 a b^2 C x-\frac{b^3 (5 A-6 C) \sin (c+d x)}{6 d}","\frac{a \left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+3 a b^2 C x-\frac{b^3 (5 A-6 C) \sin (c+d x)}{6 d}",1,"3*a*b^2*C*x + (b*(2*A*b^2 + 3*a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^3*(5*A - 6*C)*Sin[c + d*x])/(6*d) + (a*(3*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{3048, 3047, 3031, 3023, 2735, 3770}"
546,1,182,0,0.5999052,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{b \left(a^2 (4 A+6 C)+A b^2\right) \tan (c+d x)}{2 d}+\frac{a \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x","\frac{b \left(a^2 (4 A+6 C)+A b^2\right) \tan (c+d x)}{2 d}+\frac{a \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x",1,"b^3*C*x + (a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(A*b^2 + a^2*(4*A + 6*C))*Tan[c + d*x])/(2*d) + (a*(2*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,6,33,0.1818,1,"{3048, 3047, 3031, 3021, 2735, 3770}"
547,1,227,0,0.7141097,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a \left(2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{b \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(2 a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{3 A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}","\frac{a \left(2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{b \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(2 a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{3 A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}",1,"(b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(15*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (3*b*(2*A*b^2 + 5*a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a*(3*A*b^2 + 2*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (3*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,33,0.2424,1,"{3048, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
548,1,273,0,0.7862492,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{b \left(6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \left(5 a^2 (5 A+6 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{b \left(3 a^2 (4 A+5 C)+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}","\frac{b \left(6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \left(5 a^2 (5 A+6 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{b \left(3 a^2 (4 A+5 C)+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}",1,"(a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (b*(5*b^2*(2*A + 3*C) + 6*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (b*(A*b^2 + 3*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(6*A*b^2 + 5*a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",9,9,33,0.2727,1,"{3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
549,1,345,0,0.8633207,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{\left(84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right) \sin (c+d x)}{105 d}+\frac{\left(2 a^2 C+b^2 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{a b \left(6 a^2 C+126 A b^2+103 b^2 C\right) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{\left(3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac{a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d}","\frac{\left(84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right) \sin (c+d x)}{105 d}+\frac{\left(2 a^2 C+b^2 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{a b \left(6 a^2 C+126 A b^2+103 b^2 C\right) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{\left(3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac{a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d}",1,"(a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*x)/4 + ((35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + (a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a*b*(126*A*b^2 + 6*a^2*C + 103*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(210*d) + ((2*a^2*C + b^2*(7*A + 6*C))*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(21*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)","A",6,5,31,0.1613,1,"{3050, 3049, 3033, 3023, 2734}"
550,1,301,0,0.5281483,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","-\frac{a \left(-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right) \sin (c+d x)}{60 b d}-\frac{\left(4 a^2 C-5 b^2 (6 A+5 C)\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{a \left(-4 a^2 C+70 A b^2+53 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left(-2 a^2 b^2 (130 A+89 C)+8 a^4 C-15 b^4 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}","-\frac{a \left(-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right) \sin (c+d x)}{60 b d}-\frac{\left(4 a^2 C-5 b^2 (6 A+5 C)\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{a \left(-4 a^2 C+70 A b^2+53 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left(-2 a^2 b^2 (130 A+89 C)+8 a^4 C-15 b^4 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}",1,"((8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*x)/16 - (a*(4*a^4*C - 32*b^4*(5*A + 4*C) - a^2*b^2*(190*A + 121*C))*Sin[c + d*x])/(60*b*d) - ((8*a^4*C - 15*b^4*(6*A + 5*C) - 2*a^2*b^2*(130*A + 89*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) - ((4*a^2*C - 5*b^2*(6*A + 5*C))*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)","A",5,3,25,0.1200,1,"{3024, 2753, 2734}"
551,1,227,0,0.7928005,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\left(a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{a b \left(6 a^2 C+40 A b^2+29 b^2 C\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left(3 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{1}{2} a b x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}","\frac{\left(a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{a b \left(6 a^2 C+40 A b^2+29 b^2 C\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left(3 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{1}{2} a b x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x)/2 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + ((6*a^4*C + 2*b^4*(5*A + 4*C) + a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(15*d) + (a*b*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*a^2*C + b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",7,6,31,0.1935,1,"{3050, 3049, 3033, 3023, 2735, 3770}"
552,1,229,0,0.8021256,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{a b \left(a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}-\frac{b^2 \left(a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right)+\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}","-\frac{a b \left(a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}-\frac{b^2 \left(a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right)+\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}",1,"((8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x)/8 + (4*a^3*A*b*ArcTanh[Sin[c + d*x]])/d - (a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (b^2*(a^2*(24*A - 26*C) - 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (a*b*(12*A - 7*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d","A",7,6,33,0.1818,1,"{3048, 3049, 3033, 3023, 2735, 3770}"
553,1,219,0,0.9025281,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b^2 \left(a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+2 a b x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{b^2 (15 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^2}{6 d}-\frac{a b^3 (9 A-4 C) \sin (c+d x) \cos (c+d x)}{3 d}+\frac{2 A b \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}","-\frac{b^2 \left(a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+2 a b x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{b^2 (15 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^2}{6 d}-\frac{a b^3 (9 A-4 C) \sin (c+d x) \cos (c+d x)}{3 d}+\frac{2 A b \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}",1,"2*a*b*(2*A*b^2 + (2*a^2 + b^2)*C)*x + (a^2*(12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a^2*(39*A - 34*C) - 2*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (a*b^3*(9*A - 4*C)*Cos[c + d*x]*Sin[c + d*x])/(3*d) - (b^2*(15*A - 2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,33,0.2121,1,"{3048, 3047, 3049, 3033, 3023, 2735, 3770}"
554,1,251,0,0.9622182,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{2 a b \left(a^2 (2 A+3 C)+b^2 (11 A-6 C)\right) \sin (c+d x)}{3 d}+\frac{2 a b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \left(a^2 (4 A+6 C)+3 b^2 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{\left(a^2 (2 A+3 C)+6 A b^2\right) \tan (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{1}{2} b^2 x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{2 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{3 d}","-\frac{2 a b \left(a^2 (2 A+3 C)+b^2 (11 A-6 C)\right) \sin (c+d x)}{3 d}+\frac{2 a b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \left(a^2 (4 A+6 C)+3 b^2 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{\left(a^2 (2 A+3 C)+6 A b^2\right) \tan (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{1}{2} b^2 x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{2 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"(b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*x)/2 + (2*a*b*(2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(b^2*(11*A - 6*C) + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) - (b^2*(3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) + ((6*A*b^2 + a^2*(2*A + 3*C))*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(3*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,6,33,0.1818,1,"{3048, 3047, 3033, 3023, 2735, 3770}"
555,1,246,0,0.9345651,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{b^2 \left(3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right) \sin (c+d x)}{24 d}+\frac{a b \left(a^2 (23 A+36 C)+12 A b^2\right) \tan (c+d x)}{12 d}+\frac{\left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(a^2 (3 A+4 C)+4 A b^2\right) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x","-\frac{b^2 \left(3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right) \sin (c+d x)}{24 d}+\frac{a b \left(a^2 (23 A+36 C)+12 A b^2\right) \tan (c+d x)}{12 d}+\frac{\left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(a^2 (3 A+4 C)+4 A b^2\right) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x",1,"4*a*b^3*C*x + ((8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x])/(24*d) + (a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Tan[c + d*x])/(12*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,6,33,0.1818,1,"{3048, 3047, 3031, 3023, 2735, 3770}"
556,1,250,0,0.9022989,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\left(a^2 b^2 (56 A+85 C)+2 a^4 (4 A+5 C)+6 A b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left(a^2 (29 A+40 C)+6 A b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left(a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x","\frac{\left(a^2 b^2 (56 A+85 C)+2 a^4 (4 A+5 C)+6 A b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left(a^2 (29 A+40 C)+6 A b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left(a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x",1,"b^4*C*x + (a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*A*b^4 + 2*a^4*(4*A + 5*C) + a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(15*d) + (a*b*(6*A*b^2 + a^2*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*A*b^2 + a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,6,33,0.1818,1,"{3048, 3047, 3031, 3021, 2735, 3770}"
557,1,307,0,1.1245298,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{4 a b \left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{\left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b \left(a^2 (39 A+50 C)+4 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{\left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\left(5 a^2 (5 A+6 C)+12 A b^2\right) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac{2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}","\frac{4 a b \left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{\left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b \left(a^2 (39 A+50 C)+4 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{\left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\left(5 a^2 (5 A+6 C)+12 A b^2\right) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac{2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}",1,"((8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a*b*(5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((12*A*b^2 + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",9,8,33,0.2424,1,"{3048, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
558,1,355,0,1.2368256,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{\left(84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+35 b^4 (2 A+3 C)\right) \tan (c+d x)}{105 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \left(a^2 (103 A+126 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{210 d}+\frac{\left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right) \tan (c+d x) \sec ^2(c+d x)}{105 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{\left(a^2 (6 A+7 C)+2 A b^2\right) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{21 d}","\frac{\left(84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+35 b^4 (2 A+3 C)\right) \tan (c+d x)}{105 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \left(a^2 (103 A+126 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{210 d}+\frac{\left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right) \tan (c+d x) \sec ^2(c+d x)}{105 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{\left(a^2 (6 A+7 C)+2 A b^2\right) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{21 d}",1,"(a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(4*d) + ((35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[c + d*x])/(105*d) + (a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(4*d) + ((4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sec[c + d*x]^2*Tan[c + d*x])/(105*d) + (a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Sec[c + d*x]^3*Tan[c + d*x])/(210*d) + ((2*A*b^2 + a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(21*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)","A",10,9,33,0.2727,1,"{3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
559,1,183,0,0.3232107,"\int (a+b \cos (c+d x))^3 \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^3*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{b \left(-32 a^2 b^2+83 a^4-16 b^4\right) \sin (c+d x)}{30 d}+\frac{b \left(23 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{a b^2 \left(106 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(8 a^2 b^2+8 a^4-9 b^4\right)-\frac{b \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}","\frac{b \left(-32 a^2 b^2+83 a^4-16 b^4\right) \sin (c+d x)}{30 d}+\frac{b \left(23 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{a b^2 \left(106 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(8 a^2 b^2+8 a^4-9 b^4\right)-\frac{b \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}",1,"(a*(8*a^4 + 8*a^2*b^2 - 9*b^4)*x)/8 + (b*(83*a^4 - 32*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*d) + (a*b^2*(106*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(23*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (a*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) - (b*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",5,3,30,0.1000,1,"{3016, 2753, 2734}"
560,1,129,0,0.201382,"\int (a+b \cos (c+d x))^2 \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^2*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{a b \left(13 a^2-8 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(14 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^4-3 b^4\right)-\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}","\frac{a b \left(13 a^2-8 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(14 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^4-3 b^4\right)-\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}",1,"((8*a^4 - 3*b^4)*x)/8 + (a*b*(13*a^2 - 8*b^2)*Sin[c + d*x])/(6*d) + (b^2*(14*a^2 - 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",4,3,30,0.1000,1,"{3016, 2753, 2734}"
561,1,92,0,0.1124244,"\int (a+b \cos (c+d x)) \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{2 b \left(2 a^2-b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2-b^2\right)+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{2 b \left(2 a^2-b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2-b^2\right)+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(a*(2*a^2 - b^2)*x)/2 + (2*b*(2*a^2 - b^2)*Sin[c + d*x])/(3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",3,3,28,0.1071,1,"{3016, 2753, 2734}"
562,1,233,0,0.7864226,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{a \left(3 a^2 C+3 A b^2+2 b^2 C\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(4 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(4 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right)}{8 b^5}-\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}","-\frac{a \left(3 a^2 C+3 A b^2+2 b^2 C\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(4 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(4 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right)}{8 b^5}-\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"((8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x)/(8*b^5) - (2*a^3*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - (a*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^4*d) + ((4*a^2*C + b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",7,6,33,0.1818,1,"{3050, 3049, 3023, 2735, 2659, 205}"
563,1,177,0,0.4739295,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b^3 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \tan ^{-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 x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)}{2 b^4}-\frac{a C \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}","\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b^3 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \tan ^{-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 x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)}{2 b^4}-\frac{a C \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"-(a*(2*A*b^2 + (2*a^2 + b^2)*C)*x)/(2*b^4) + (2*a^2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2*C + b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b^3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",6,6,33,0.1818,1,"{3050, 3049, 3023, 2735, 2659, 205}"
564,1,126,0,0.2583645,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 a \left(a^2 C+A b^2\right) \tan ^{-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{x \left(\frac{2 a^2 C}{b^2}+2 A+C\right)}{2 b}-\frac{a C \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{2 a \left(a^2 C+A b^2\right) \tan ^{-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{x \left(2 a^2 C+b^2 (2 A+C)\right)}{2 b^3}-\frac{a C \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"((2*A + C + (2*a^2*C)/b^2)*x)/(2*b) - (2*a*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*C*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,31,0.1613,1,"{3050, 3023, 2735, 2659, 205}"
565,1,86,0,0.1257375,"\int \frac{A+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C x}{b^2}+\frac{C \sin (c+d x)}{b d}","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C x}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"-((a*C*x)/b^2) + (2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)","A",4,4,25,0.1600,1,"{3024, 2735, 2659, 205}"
566,1,88,0,0.1309406,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}","-\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}",1,"(C*x)/b - (2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d) + (A*ArcTanh[Sin[c + d*x]])/(a*d)","A",4,4,31,0.1290,1,"{3058, 2659, 205, 3770}"
567,1,95,0,0.2250401,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{A b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{A b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}",1,"(2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (A*b*ArcTanh[Sin[c + d*x]])/(a^2*d) + (A*Tan[c + d*x])/(a*d)","A",5,5,33,0.1515,1,"{3056, 3001, 3770, 2659, 205}"
568,1,137,0,0.4737365,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","-\frac{2 b \left(a^2 C+A b^2\right) \tan ^{-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{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{A b \tan (c+d x)}{a^2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 b \left(a^2 C+A b^2\right) \tan ^{-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{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{A b \tan (c+d x)}{a^2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(-2*b*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (A*b*Tan[c + d*x])/(a^2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",6,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
569,1,184,0,0.740514,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{2 b^2 \left(a^2 C+A b^2\right) \tan ^{-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{\left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{A b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}","\frac{2 b^2 \left(a^2 C+A b^2\right) \tan ^{-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{\left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{A b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"(2*b^2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*a^3*d) - (A*b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
570,1,332,0,1.117829,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C+4 a^4 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2 C+3 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{a x \left(C \left(4 a^2+b^2\right)+2 A b^2\right)}{b^5}","\frac{\left(a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C+4 a^4 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2 C+3 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{a x \left(C \left(4 a^2+b^2\right)+2 A b^2\right)}{b^5}",1,"-((a*(2*A*b^2 + (4*a^2 + b^2)*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((a^2*b^2*(6*A - 7*C) + 12*a^4*C - b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3048, 3049, 3023, 2735, 2659, 205}"
571,1,262,0,0.6896762,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{a \left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(a^2 A b^2-4 a^2 b^2 C+3 a^4 C-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)}{2 b^4}","-\frac{a \left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(a^2 A b^2-4 a^2 b^2 C+3 a^4 C-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)}{2 b^4}",1,"((2*A*b^2 + (6*a^2 + b^2)*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3048, 3049, 3023, 2735, 2659, 205}"
572,1,144,0,0.3463463,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(3 a^2 b^2 C-2 a^4 C+A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d}","-\frac{2 \left(3 a^2 b^2 C-2 a^4 C+A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"(-2*a*C*x)/b^3 - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3032, 3023, 2735, 2659, 205}"
573,1,126,0,0.1963378,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 a \left(a^2 (-C)+A b^2+2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}","\frac{2 a \left(a^2 (-C)+A b^2+2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*x)/b^2 + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,25,0.1600,1,"{3022, 2735, 2659, 205}"
574,1,134,0,0.3318431,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b \left(2 a^2 A+a^2 C-A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}","-\frac{2 b \left(2 a^2 A+a^2 C-A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"(-2*b*(2*a^2*A - A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3056, 3001, 3770, 2659, 205}"
575,1,180,0,0.5819779,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(3 a^2 A b^2+a^4 C-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(2 A b^2-a^2 (A-C)\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 A b \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{2 \left(3 a^2 A b^2+a^4 C-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(2 A b^2-a^2 (A-C)\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 A b \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - (2*A*b*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((2*A*b^2 - a^2*(A - C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
576,1,265,0,1.0638681,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C+2 a^4 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b \left(3 A b^2-a^2 (2 A-C)\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\left(3 A b^2-a^2 (A-2 C)\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C+2 a^4 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b \left(3 A b^2-a^2 (2 A-C)\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\left(3 A b^2-a^2 (A-2 C)\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(-2*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((6*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + (b*(3*A*b^2 - a^2*(2*A - C))*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - a^2*(A - 2*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
577,1,335,0,1.4696152,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C+3 a^4 C-4 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+12 A b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2 (A+2 C)+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left(4 A b^2-a^2 (A-3 C)\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b \left(2 A b^2-a^2 (A-C)\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C+3 a^4 C-4 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+12 A b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2 (A+2 C)+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left(4 A b^2-a^2 (A-3 C)\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b \left(2 A b^2-a^2 (A-C)\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(4*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(a^5*d) - ((12*A*b^4 - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) + (b*(2*A*b^2 - a^2*(A - C))*Sec[c + d*x]*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((4*A*b^2 - a^2*(A - 3*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
578,1,372,0,1.4892669,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(a^2 b^2 (2 A-21 C)+12 a^4 C-b^4 (5 A-6 C)\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C+6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(7 a^2 b^2 C-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (A-10 C)+6 a^4 C-b^4 (4 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)}{2 b^5}","-\frac{a \left(a^2 b^2 (2 A-21 C)+12 a^4 C-b^4 (5 A-6 C)\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C+6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(7 a^2 b^2 C-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (A-10 C)+6 a^4 C-b^4 (4 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)}{2 b^5}",1,"((2*A*b^2 + (12*a^2 + b^2)*C)*x)/(2*b^5) - (a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (a*(a^2*b^2*(2*A - 21*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*b^2*(A - 10*C) - b^4*(4*A - C) + 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3048, 3047, 3049, 3023, 2735, 2659, 205}"
579,1,262,0,0.8413999,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 b^4 (A+12 C)-15 a^4 b^2 C+6 a^6 C+2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(a^2 b^2 (A+6 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{3 a C x}{b^4}","\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 b^4 (A+12 C)-15 a^4 b^2 C+6 a^6 C+2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(a^2 b^2 (A+6 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{3 a C x}{b^4}",1,"(-3*a*C*x)/b^4 + ((2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3048, 3031, 3023, 2735, 2659, 205}"
580,1,203,0,0.4644766,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(C \left(-5 a^2 b^2+2 a^4+6 b^4\right)+3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^2 b^2 (A+6 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{C x}{b^3}","-\frac{a \left(C \left(-5 a^2 b^2+2 a^4+6 b^4\right)+3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^2 b^2 (A+6 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{C x}{b^3}",1,"(C*x)/b^3 - (a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3032, 3021, 2735, 2659, 205}"
581,1,177,0,0.2625128,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2 (2 A+C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a \left(a^2 (-C)+3 A b^2+4 b^2 C\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(a^2 (2 A+C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a \left(a^2 (-C)+3 A b^2+4 b^2 C\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((a^2*(2*A + C) + b^2*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,25,0.2000,1,"{3022, 2754, 12, 2659, 205}"
582,1,211,0,0.6488924,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{b \left(5 a^2 A b^2-3 a^4 (2 A+C)-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (5 A+2 C)+a^4 (-C)+2 A b^4\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{b \left(5 a^2 A b^2-3 a^4 (2 A+C)-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (5 A+2 C)+a^4 (-C)+2 A b^4\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(b*(5*a^2*A*b^2 - 2*A*b^4 - 3*a^4*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((2*A*b^4 - a^4*C - a^2*b^2*(5*A + 2*C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{3056, 3055, 3001, 3770, 2659, 205}"
583,1,275,0,1.1893726,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-2 a^6 C-6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(11 a^2 A b^2+a^4 (-(2 A-3 C))-6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (6 A+C)-2 a^4 C+3 A b^4\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 A b \tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-2 a^6 C-6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(11 a^2 A b^2+a^4 (-(2 A-3 C))-6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (6 A+C)-2 a^4 C+3 A b^4\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 A b \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-(((15*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d)) - (3*A*b*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((11*a^2*A*b^2 - 6*A*b^4 - a^4*(2*A - 3*C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(6*A + C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
584,1,378,0,1.8758381,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right) \tan ^{-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(-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(7 a^2 A b^2+3 a^4 C-4 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right) \tan ^{-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(-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(7 a^2 A b^2+3 a^4 C-4 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
585,1,514,0,2.2104299,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{a \left(a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+60 a^6 C+2 b^6 (13 A-12 C)\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)^3}+\frac{\left(-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C+8 a A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^4(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 b^2 (A+10 C)-5 a^4 C+4 A b^4\right) \sin (c+d x) \cos ^3(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^6 C+12 A b^6\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+10 a^6 C+b^6 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3}+\frac{x \left(C \left(20 a^2+b^2\right)+2 A b^2\right)}{2 b^6}","-\frac{a \left(a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+60 a^6 C+2 b^6 (13 A-12 C)\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)^3}+\frac{\left(-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C+8 a A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^4(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 b^2 (A+10 C)-5 a^4 C+4 A b^4\right) \sin (c+d x) \cos ^3(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^6 C+12 A b^6\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+10 a^6 C+b^6 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3}+\frac{x \left(C \left(20 a^2+b^2\right)+2 A b^2\right)}{2 b^6}",1,"((2*A*b^2 + (20*a^2 + b^2)*C)*x)/(2*b^6) + ((8*a*A*b^8 - a^7*b^2*(2*A - 69*C) + 7*a^5*b^4*(A - 12*C) - 8*a^3*b^6*(A - 5*C) - 20*a^9*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2 - b^2)^3*d) - (a*(a^4*b^2*(6*A - 167*C) - a^2*b^4*(17*A - 146*C) + 2*b^6*(13*A - 12*C) + 60*a^6*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*b^2*(A - 27*C) - a^2*b^4*(2*A - 23*C) + b^6*(6*A - C) + 10*a^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((4*A*b^4 - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,7,33,0.2121,1,"{3048, 3047, 3049, 3023, 2735, 2659, 205}"
586,1,369,0,1.5789243,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(5 A b^4-C \left(-23 a^2 b^2+12 a^4+6 b^4\right)\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 C+2 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 C+2 A b^6\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{4 a C x}{b^5}","-\frac{\left(5 A b^4-C \left(-23 a^2 b^2+12 a^4+6 b^4\right)\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 C+2 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 C+2 A b^6\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{4 a C x}{b^5}",1,"(-4*a*C*x)/b^5 - ((2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (12*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3048, 3047, 3031, 3023, 2735, 2659, 205}"
587,1,304,0,1.0844357,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{a \left(a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 C+4 b^6 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 C+4 A b^6\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \left(a^2 b^2 (3 A+8 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{C x}{b^4}","\frac{a \left(a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 C+4 b^6 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 C+4 A b^6\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \left(a^2 b^2 (3 A+8 C)-3 a^4 C+2 A b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{C x}{b^4}",1,"(C*x)/b^4 + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*b^6*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((4*A*b^6 + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3048, 3031, 3021, 2735, 2659, 205}"
588,1,261,0,0.5665253,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","-\frac{b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(a^2 b^2 (2 A-5 C)+2 a^4 C+b^4 (13 A+18 C)\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"-((b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b^2*(2*A - 5*C) + 2*a^4*C + b^4*(13*A + 18*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{3032, 3021, 2754, 12, 2659, 205}"
589,1,252,0,0.4631399,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{a \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-a^2 b^2 (11 A+10 C)+a^4 C-2 b^4 (2 A+3 C)\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \left(a^2 (-C)+5 A b^2+6 b^2 C\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{a \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-a^2 b^2 (11 A+10 C)+a^4 C-2 b^4 (2 A+3 C)\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \left(a^2 (-C)+5 A b^2+6 b^2 C\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"(a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,5,25,0.2000,1,"{3022, 2754, 12, 2659, 205}"
590,1,301,0,1.3016687,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","-\frac{b \left(-a^4 b^2 (8 A-C)+7 a^2 A b^4+4 a^6 (2 A+C)-2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-2 a^6 C-6 A b^6\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (8 A+3 C)-2 a^4 C+3 A b^4\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{b \left(-a^4 b^2 (8 A-C)+7 a^2 A b^4+4 a^6 (2 A+C)-2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-2 a^6 C-6 A b^6\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (8 A+3 C)-2 a^4 C+3 A b^4\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-((b*(7*a^2*A*b^4 - 2*A*b^6 - a^4*b^2*(8*A - C) + 4*a^6*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,6,31,0.1935,1,"{3056, 3055, 3001, 3770, 2659, 205}"
591,1,376,0,2.1274076,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-4 A b^6\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (9 A+2 C)-3 a^4 C+4 A b^4\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{4 A b \tanh ^{-1}(\sin (c+d x))}{a^5 d}","-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-4 A b^6\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (9 A+2 C)-3 a^4 C+4 A b^4\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{4 A b \tanh ^{-1}(\sin (c+d x))}{a^5 d}",1,"-(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (4*A*b*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
592,1,522,0,2.658623,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","\frac{\left(-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{b \left(a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 (A+2 C)+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{\left(-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{b \left(a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 (A+2 C)+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(12*A - C) - 8*a^8*b*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((20*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((20*A*b^6 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",9,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
593,1,193,0,0.6147955,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{a \left(3 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}+\frac{2 a^3 \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^5 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^5}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}","\frac{a \left(3 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}+\frac{2 a^3 \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^5 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^5}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"-((8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^5) + (2*a^3*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(b^5*d) + (a*(3*a^2 - b^2)*Sin[c + d*x])/(3*b^4*d) - ((4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",7,6,33,0.1818,1,"{3050, 3049, 3023, 2735, 2659, 205}"
594,1,150,0,0.3896282,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{\left(3 a^2-b^2\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^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^4 d}+\frac{a x \left(2 a^2-b^2\right)}{2 b^4}+\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}","-\frac{\left(3 a^2-b^2\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^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^4 d}+\frac{a x \left(2 a^2-b^2\right)}{2 b^4}+\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(a*(2*a^2 - b^2)*x)/(2*b^4) - (2*a^2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(b^4*d) - ((3*a^2 - b^2)*Sin[c + d*x])/(3*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",6,6,33,0.1818,1,"{3050, 3049, 3023, 2735, 2659, 205}"
595,1,109,0,0.2002136,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{x \left(2 a^2-b^2\right)}{2 b^3}+\frac{a \sin (c+d x)}{b^2 d}+\frac{2 a \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^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{x \left(2 a^2-b^2\right)}{2 b^3}+\frac{a \sin (c+d x)}{b^2 d}+\frac{2 a \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^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"-((2*a^2 - b^2)*x)/(2*b^3) + (2*a*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(b^3*d) + (a*Sin[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,31,0.1613,1,"{3050, 3023, 2735, 2659, 205}"
596,1,73,0,0.1049906,"\int \frac{1-\cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),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{\sin (c+d x)}{b d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d}+\frac{a x}{b^2}-\frac{\sin (c+d x)}{b d}",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) - Sin[c + d*x]/(b*d)","A",4,4,25,0.1600,1,"{3024, 2735, 2659, 205}"
597,1,76,0,0.1165543,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),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)}{a b d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{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)}{a b d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{b}",1,"-(x/b) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*b*d) + ArcTanh[Sin[c + d*x]]/(a*d)","A",4,4,31,0.1290,1,"{3058, 2659, 205, 3770}"
598,1,82,0,0.2064429,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),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)}{a^2 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}",1,"(-2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*d) - (b*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a*d)","A",5,5,33,0.1515,1,"{3056, 3001, 3770, 2659, 205}"
599,1,117,0,0.3743386,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","-\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{2 b \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)}{a^3 d}-\frac{b \tan (c+d x)}{a^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{2 b \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)}{a^3 d}-\frac{b \tan (c+d x)}{a^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"(2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*d) - ((a^2 - 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*Tan[c + d*x])/(a^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",6,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
600,1,155,0,0.5697023,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","-\frac{2 b^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)}{a^4 d}-\frac{\left(a^2-3 b^2\right) \tan (c+d x)}{3 a^3 d}+\frac{b \left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}","-\frac{2 b^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)}{a^4 d}-\frac{\left(a^2-3 b^2\right) \tan (c+d x)}{3 a^3 d}+\frac{b \left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"(-2*Sqrt[a - b]*b^2*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*d) + (b*(a^2 - 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - ((a^2 - 3*b^2)*Tan[c + d*x])/(3*a^3*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
601,1,237,0,0.8984754,"\int \frac{\cos ^4(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{a \left(15 a^2-2 b^2\right) \sin (c+d x)}{3 b^5 d}+\frac{2 a^3 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left(20 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac{x \left(-12 a^2 b^2+40 a^4-b^4\right)}{8 b^6}+\frac{5 a \sin (c+d x) \cos ^2(c+d x)}{3 b^3 d}+\frac{\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 b^2 d}","\frac{a \left(15 a^2-2 b^2\right) \sin (c+d x)}{3 b^5 d}+\frac{2 a^3 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left(20 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac{x \left(-12 a^2 b^2+40 a^4-b^4\right)}{8 b^6}+\frac{5 a \sin (c+d x) \cos ^2(c+d x)}{3 b^3 d}+\frac{\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 b^2 d}",1,"-((40*a^4 - 12*a^2*b^2 - b^4)*x)/(8*b^6) + (2*a^3*(5*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*d) + (a*(15*a^2 - 2*b^2)*Sin[c + d*x])/(3*b^5*d) - ((20*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + (5*a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^3*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))","A",8,7,33,0.2121,1,"{3048, 3050, 3049, 3023, 2735, 2659, 205}"
602,1,189,0,0.6164765,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\left(12 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^2 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{a x \left(4 a^2-b^2\right)}{b^5}+\frac{2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}","-\frac{\left(12 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^2 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{a x \left(4 a^2-b^2\right)}{b^5}+\frac{2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}",1,"(a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - ((12*a^2 - b^2)*Sin[c + d*x])/(3*b^4*d) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3048, 3050, 3049, 3023, 2735, 2659, 205}"
603,1,154,0,0.3972349,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{2 a \left(3 a^2-2 b^2\right) \tan ^{-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{x \left(6 a^2-b^2\right)}{2 b^4}+\frac{3 a \sin (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{b d (a+b \cos (c+d x))}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 b^2 d}","\frac{2 a \left(3 a^2-2 b^2\right) \tan ^{-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{x \left(6 a^2-b^2\right)}{2 b^4}+\frac{3 a \sin (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{b d (a+b \cos (c+d x))}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 b^2 d}",1,"-((6*a^2 - b^2)*x)/(2*b^4) + (2*a*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + (3*a*Sin[c + d*x])/(b^3*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3048, 3050, 3023, 2735, 2659, 205}"
604,1,112,0,0.2582684,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(2 a^2-b^2\right) \tan ^{-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 \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}+\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d}","-\frac{2 \left(2 a^2-b^2\right) \tan ^{-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 \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}+\frac{2 a x}{b^3}-\frac{\sin (c+d x)}{b^2 d}",1,"(2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - Sin[c + d*x]/(b^2*d) - (a*Sin[c + d*x])/(b^2*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3032, 3023, 2735, 2659, 205}"
605,1,85,0,0.1183239,"\int \frac{1-\cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 a \tan ^{-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{\sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac{x}{b^2}","\frac{2 a \tan ^{-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{\sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac{x}{b^2}",1,"-(x/b^2) + (2*a*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Sin[c + d*x]/(b*d*(a + b*Cos[c + d*x]))","A",5,5,25,0.2000,1,"{3022, 12, 2735, 2659, 205}"
606,1,94,0,0.1703114,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b \tan ^{-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{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\sin (c+d x)}{a d (a+b \cos (c+d x))}","-\frac{2 b \tan ^{-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{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\sin (c+d x)}{a d (a+b \cos (c+d x))}",1,"(-2*b*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + ArcTanh[Sin[c + d*x]]/(a^2*d) - Sin[c + d*x]/(a*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{3056, 12, 2747, 3770, 2659, 205}"
607,1,118,0,0.3991453,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-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{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{2 \tan (c+d x)}{a^2 d}-\frac{\tan (c+d x)}{a d (a+b \cos (c+d x))}","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-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{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{2 \tan (c+d x)}{a^2 d}-\frac{\tan (c+d x)}{a d (a+b \cos (c+d x))}",1,"(-2*(a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - (2*b*ArcTanh[Sin[c + d*x]])/(a^3*d) + (2*Tan[c + d*x])/(a^2*d) - Tan[c + d*x]/(a*d*(a + b*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{3056, 3001, 3770, 2659, 205}"
608,1,160,0,0.6724921,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b \left(2 a^2-3 b^2\right) \tan ^{-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{\left(a^2-6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{3 b \tan (c+d x)}{a^3 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{a d (a+b \cos (c+d x))}","\frac{2 b \left(2 a^2-3 b^2\right) \tan ^{-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{\left(a^2-6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{3 b \tan (c+d x)}{a^3 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{a d (a+b \cos (c+d x))}",1,"(2*b*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((a^2 - 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (3*b*Tan[c + d*x])/(a^3*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
609,1,195,0,0.9162848,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-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{\left(a^2-12 b^2\right) \tan (c+d x)}{3 a^4 d}+\frac{b \left(a^2-4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{2 b \tan (c+d x) \sec (c+d x)}{a^3 d}+\frac{4 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{a d (a+b \cos (c+d x))}","-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-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{\left(a^2-12 b^2\right) \tan (c+d x)}{3 a^4 d}+\frac{b \left(a^2-4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{2 b \tan (c+d x) \sec (c+d x)}{a^3 d}+\frac{4 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{a d (a+b \cos (c+d x))}",1,"(-2*b^2*(3*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) + (b*(a^2 - 4*b^2)*ArcTanh[Sin[c + d*x]])/(a^5*d) - ((a^2 - 12*b^2)*Tan[c + d*x])/(3*a^4*d) - (2*b*Sec[c + d*x]*Tan[c + d*x])/(a^3*d) + (4*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
610,1,326,0,1.0467084,"\int \frac{\cos ^4(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-59 a^2 b^2+60 a^4+2 b^4\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)}-\frac{a^2 \left(-33 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(5 a^2-4 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(20 a^2-17 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a x \left(20 a^2-3 b^2\right)}{2 b^6}+\frac{\sin (c+d x) \cos ^4(c+d x)}{2 b d (a+b \cos (c+d x))^2}","-\frac{\left(-59 a^2 b^2+60 a^4+2 b^4\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)}-\frac{a^2 \left(-33 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(5 a^2-4 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(20 a^2-17 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a x \left(20 a^2-3 b^2\right)}{2 b^6}+\frac{\sin (c+d x) \cos ^4(c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"(a*(20*a^2 - 3*b^2)*x)/(2*b^6) - (a^2*(20*a^4 - 33*a^2*b^2 + 12*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^6*(a + b)^(3/2)*d) - ((60*a^4 - 59*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)*d) + (a*(10*a^2 - 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((20*a^2 - 17*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) + ((5*a^2 - 4*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3048, 3049, 3023, 2735, 2659, 205}"
611,1,268,0,0.7506208,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{a \left(12 a^2-11 b^2\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-19 a^2 b^2+12 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(4 a^2-3 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{x \left(12 a^2-b^2\right)}{2 b^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2}","\frac{a \left(12 a^2-11 b^2\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-19 a^2 b^2+12 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(4 a^2-3 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{x \left(12 a^2-b^2\right)}{2 b^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"-((12*a^2 - b^2)*x)/(2*b^5) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + (a*(12*a^2 - 11*b^2)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((6*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) + ((4*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3048, 3049, 3023, 2735, 2659, 205}"
612,1,182,0,0.4583769,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-9 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-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 \left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{3 a x}{b^4}+\frac{\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac{3 \sin (c+d x)}{2 b^3 d}","-\frac{\left(-9 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-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 \left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{3 a x}{b^4}+\frac{\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac{3 \sin (c+d x)}{2 b^3 d}",1,"(3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (3*Sin[c + d*x])/(2*b^3*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3048, 3032, 3023, 2735, 2659, 205}"
613,1,149,0,0.2916691,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{a \left(2 a^2-3 b^2\right) \tan ^{-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{\left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{a \sin (c+d x)}{2 b^2 d (a+b \cos (c+d x))^2}-\frac{x}{b^3}","\frac{a \left(2 a^2-3 b^2\right) \tan ^{-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{\left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{a \sin (c+d x)}{2 b^2 d (a+b \cos (c+d x))^2}-\frac{x}{b^3}",1,"-(x/b^3) + (a*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) - (a*Sin[c + d*x])/(2*b^2*d*(a + b*Cos[c + d*x])^2) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{3032, 3021, 2735, 2659, 205}"
614,1,117,0,0.1298795,"\int \frac{1-\cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{a \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tan ^{-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{\sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}","-\frac{a \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tan ^{-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{\sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]]/((a - b)^(3/2)*(a + b)^(3/2)*d) + Sin[c + d*x]/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,5,25,0.2000,1,"{3022, 12, 2754, 2659, 205}"
615,1,155,0,0.4117327,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","-\frac{b \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\sin (c+d x)}{2 a d (a+b \cos (c+d x))^2}","-\frac{b \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\sin (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"-((b*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(2*a*d*(a + b*Cos[c + d*x])^2) - ((a^2 - 2*b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,5,31,0.1613,1,"{3056, 3001, 3770, 2659, 205}"
616,1,204,0,0.7353748,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\left(2 a^2-3 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{\tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\left(2 a^2-3 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{\tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"-(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) - (3*b*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((5*a^2 - 6*b^2)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - Tan[c + d*x]/(2*a*d*(a + b*Cos[c + d*x])^2) - ((2*a^2 - 3*b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
617,1,271,0,1.0476592,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{b \left(-19 a^2 b^2+6 a^4+12 b^4\right) \tan ^{-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 \left(11 a^2-12 b^2\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)}-\frac{\left(a^2-12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\left(3 a^2-4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^2}","\frac{b \left(-19 a^2 b^2+6 a^4+12 b^4\right) \tan ^{-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 \left(11 a^2-12 b^2\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)}-\frac{\left(a^2-12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\left(3 a^2-4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((a^2 - 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (b*(11*a^2 - 12*b^2)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)*d) + ((5*a^2 - 6*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^2) - ((3*a^2 - 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
618,1,335,0,1.3977825,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^3,x]","-\frac{b^2 \left(-33 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(-59 a^2 b^2+2 a^4+60 b^4\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)}+\frac{b \left(3 a^2-20 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left(17 a^2-20 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 a^3 d \left(a^2-b^2\right)}-\frac{b \left(9 a^2-10 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)}-\frac{\left(4 a^2-5 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 a d (a+b \cos (c+d x))^2}","-\frac{b^2 \left(-33 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(-59 a^2 b^2+2 a^4+60 b^4\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)}+\frac{b \left(3 a^2-20 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left(17 a^2-20 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 a^3 d \left(a^2-b^2\right)}-\frac{b \left(9 a^2-10 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)}-\frac{\left(4 a^2-5 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"-((b^2*(12*a^4 - 33*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (b*(3*a^2 - 20*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((2*a^4 - 59*a^2*b^2 + 60*b^4)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)*d) - (b*(9*a^2 - 10*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)*d) + ((17*a^2 - 20*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a^3*(a^2 - b^2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^2) - ((4*a^2 - 5*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",9,6,33,0.1818,1,"{3056, 3055, 3001, 3770, 2659, 205}"
619,1,16,0,0.045377,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","a x-\frac{b \sin (c+d x)}{d}","a x-\frac{b \sin (c+d x)}{d}",1,"a*x - (b*Sin[c + d*x])/d","A",3,2,30,0.06667,1,"{3016, 2637}"
620,1,54,0,0.0996256,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{4 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}-x","\frac{4 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}-x",1,"-x + (4*a*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",4,4,30,0.1333,1,"{3016, 2735, 2659, 205}"
621,1,93,0,0.1169771,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{2 \left(a^2+b^2\right) \tan ^{-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{2 a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 \left(a^2+b^2\right) \tan ^{-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{2 a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*(a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (2*a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,30,0.1667,1,"{3016, 2754, 12, 2659, 205}"
622,1,140,0,0.2325846,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{2 a \left(a^2+2 b^2\right) \tan ^{-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{b \left(2 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{2 a \left(a^2+2 b^2\right) \tan ^{-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{b \left(2 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"(2*a*(a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(2*a^2 + b^2)*Sin[c + d*x])/((a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,5,30,0.1667,1,"{3016, 2754, 12, 2659, 205}"
623,1,364,0,0.8055284,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}-\frac{4 a \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}+\frac{4 a \left(a^2-b^2\right) \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(6 a^2 b^2 (7 A+4 C)+16 a^4 C-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}","\frac{2 \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}-\frac{4 a \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}+\frac{4 a \left(a^2-b^2\right) \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(6 a^2 b^2 (7 A+4 C)+16 a^4 C-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}",1,"(-2*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*(21*A*b^2 + 8*a^2*C + 18*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(21*A*b^2 + 8*a^2*C + 18*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(24*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) - (4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)","A",9,9,35,0.2571,1,"{3050, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
624,1,291,0,0.4797501,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(8 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}","\frac{2 \left(8 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(2*a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 + 8*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)","A",8,8,33,0.2424,1,"{3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
625,1,218,0,0.325578,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \left(2 a^2 C-3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}","-\frac{2 \left(2 a^2 C-3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}",1,"(-2*(2*a^2*C - 3*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,27,0.2593,1,"{3024, 2753, 2752, 2663, 2661, 2655, 2653}"
626,1,231,0,0.6470116,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{2 \left(a^2 C-b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 C-b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*a*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2*C - b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,33,0.2727,1,"{3050, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
627,1,205,0,0.6196901,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"-(((A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",9,9,35,0.2571,1,"{3048, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
628,1,277,0,0.9896185,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{\left(A b^2-4 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{b (3 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\left(A b^2-4 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{b (3 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-(A*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(3*A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^2 - 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,35,0.2857,1,"{3048, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
629,1,365,0,1.3604017,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(A b^2-8 a^2 (2 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}","-\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(A b^2-8 a^2 (2 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}",1,"((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^2*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,35,0.2857,1,"{3048, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
630,1,443,0,1.0632625,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{231 b^3 d}-\frac{4 a \left(8 a^2 C+33 A b^2+34 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{1155 b^3 d}-\frac{2 \left(6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1155 b^3 d}+\frac{2 \left(a^2-b^2\right) \left(6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(3 a^2 b^2 (11 A+6 C)+8 a^4 C-b^4 (451 A+348 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{33 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}","\frac{2 \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{231 b^3 d}-\frac{4 a \left(8 a^2 C+33 A b^2+34 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{1155 b^3 d}-\frac{2 \left(6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1155 b^3 d}+\frac{2 \left(a^2-b^2\right) \left(6 a^2 b^2 (11 A+8 C)+16 a^4 C-25 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(3 a^2 b^2 (11 A+6 C)+8 a^4 C-b^4 (451 A+348 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{33 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}",1,"(-4*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(1155*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(1155*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1155*b^3*d) - (4*a*(33*A*b^2 + 8*a^2*C + 34*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*b^3*d) + (2*(8*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(231*b^3*d) - (4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(33*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)","A",10,9,35,0.2571,1,"{3050, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
631,1,356,0,0.6529027,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(8 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(a^2-b^2\right) \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}","\frac{2 \left(8 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(a^2-b^2\right) \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(2*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)","A",9,8,33,0.2424,1,"{3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
632,1,285,0,0.4691811,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \left(6 a^2 C-5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(-3 a^2 C+70 A b^2+41 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}","-\frac{2 \left(6 a^2 C-5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(-3 a^2 C+70 A b^2+41 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}",1,"(4*a*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2*C - 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,7,27,0.2593,1,"{3024, 2753, 2752, 2663, 2661, 2655, 2653}"
633,1,281,0,0.9233101,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 a \left(5 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","\frac{2 a \left(5 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(a^2*C + b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(5*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*a*(5*A*b^2 - (a^2 - b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(5*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",10,10,33,0.3030,1,"{3050, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
634,1,270,0,0.9322679,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\left(a^2 (3 A-2 C)+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{a (3 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}+\frac{3 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{\left(a^2 (3 A-2 C)+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{a (3 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}+\frac{3 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"-(a*(3*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((a^2*(3*A - 2*C) + 2*b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (3*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",10,10,35,0.2857,1,"{3048, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
635,1,276,0,0.9513706,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a b (7 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a b (7 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"-(b*(5*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*b*(7*A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (3*A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,35,0.2857,1,"{3048, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
636,1,365,0,1.4412473,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(A b^2-12 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}","\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(A b^2-12 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}",1,"-((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((8*a^2*(2*A + 3*C) + b^2*(17*A + 48*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(A*b^2 - 12*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,11,35,0.3143,1,"{3048, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
637,1,436,0,1.8064832,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{64 a^2 d}-\frac{b \left(A b^2-4 a^2 (19 A+28 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (3 A+4 C)+A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 a d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{8 d}","-\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{64 a^2 d}-\frac{b \left(A b^2-4 a^2 (19 A+28 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (3 A+4 C)+A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 a d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{8 d}",1,"(b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (b*(A*b^2 - 4*a^2*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(64*a^2*d) + ((A*b^2 + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*a*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(8*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,11,35,0.3143,1,"{3048, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
638,1,523,0,1.2795254,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}-\frac{4 a \left(24 a^2 C+143 A b^2+166 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}-\frac{2 \left(10 a^2 b^2 (143 A+124 C)+240 a^4 C-539 b^4 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}-\frac{4 a \left(5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}+\frac{4 a \left(a^2-b^2\right) \left(5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 C-1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}","\frac{2 \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}-\frac{4 a \left(24 a^2 C+143 A b^2+166 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}-\frac{2 \left(10 a^2 b^2 (143 A+124 C)+240 a^4 C-539 b^4 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}-\frac{4 a \left(5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}+\frac{4 a \left(a^2-b^2\right) \left(5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 C-1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}",1,"(-2*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) - (2*(240*a^4*C - 539*b^4*(13*A + 11*C) + 10*a^2*b^2*(143*A + 124*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) - (4*a*(143*A*b^2 + 24*a^2*C + 166*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(24*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) - (12*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d)","A",11,9,35,0.2571,1,"{3050, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
639,1,435,0,0.879467,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(8 a^2 C+9 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2 C+99 A b^2+67 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}","\frac{2 \left(8 a^2 C+9 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2 C+99 A b^2+67 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(2*a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)","A",10,8,33,0.2424,1,"{3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
640,1,350,0,0.6487882,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \left(10 a^2 C-7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{4 a \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{4 a \left(a^2-b^2\right) \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}","-\frac{2 \left(10 a^2 C-7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{4 a \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{4 a \left(a^2-b^2\right) \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}",1,"(-2*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)","A",9,7,27,0.2593,1,"{3024, 2753, 2752, 2663, 2661, 2655, 2653}"
641,1,342,0,1.2230645,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \left(3 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 \left(2 a^2 b^2 (7 A-C)-3 a^4 C+b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2 C+49 A b^2+29 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}","\frac{2 \left(3 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 \left(2 a^2 b^2 (7 A-C)-3 a^4 C+b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2 C+49 A b^2+29 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*b^2*(7*A - C) - 3*a^4*C + b^4*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2*C + b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",11,10,33,0.3030,1,"{3050, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
642,1,327,0,1.2741611,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a \left(a^2 (15 A-16 C)+4 b^2 (15 A+4 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 a^2 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{a b (15 A-16 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}","\frac{a \left(a^2 (15 A-16 C)+4 b^2 (15 A+4 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 a^2 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{a b (15 A-16 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}",1,"-((a^2*(15*A - 46*C) - 6*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(a^2*(15*A - 16*C) + 4*b^2*(15*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (5*a^2*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (a*b*(15*A - 16*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d","A",11,10,35,0.2857,1,"{3048, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
643,1,329,0,1.2728219,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{b \left(a^2 (33 A+16 C)+8 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b^2 (21 A-8 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{a b (27 A-56 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 A b \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}","\frac{b \left(a^2 (33 A+16 C)+8 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b^2 (21 A-8 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{a b (27 A-56 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 A b \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}",1,"-(a*b*(27*A - 56*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(8*b^2*(3*A + C) + a^2*(33*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - (b^2*(21*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,11,35,0.3143,1,"{3048, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
644,1,363,0,1.416339,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(8 a^2 (2 A+3 C)+b^2 (59 A+96 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{5 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(8 a^2 (2 A+3 C)+b^2 (59 A+96 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{5 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}",1,"-((3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(8*a^2*(2*A + 3*C) + b^2*(59*A + 96*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*(A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,35,0.2857,1,"{3048, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
645,1,437,0,1.8745452,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{b \left(4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (3 A+4 C)+5 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}","\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{b \left(4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (3 A+4 C)+5 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}",1,"-(b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(4*a^2*(89*A + 132*C) + b^2*(133*A + 384*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^4 - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((5*A*b^2 + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,11,35,0.3143,1,"{3048, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
646,1,246,0,0.4575301,"\int (a+b \cos (c+d x))^{3/2} \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{2 b \left(41 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-66 a^2 b^2+41 a^4+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(73 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{4 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}","\frac{2 b \left(41 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-66 a^2 b^2+41 a^4+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(73 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{4 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}",1,"(4*a*(73*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(41*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (4*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",9,7,32,0.2188,1,"{3016, 2753, 2752, 2663, 2661, 2655, 2653}"
647,1,197,0,0.3376216,"\int \sqrt{a+b \cos (c+d x)} \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(a^2 - b^2*Cos[c + d*x]^2),x]","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(17 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{4 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(17 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{4 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(17*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",8,7,32,0.2188,1,"{3016, 2753, 2752, 2663, 2661, 2655, 2653}"
648,1,378,0,0.9266818,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(48 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{4 a \left(32 a^2 C+42 A b^2+31 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^4 d}-\frac{2 a \left(4 a^2 b^2 (42 A+19 C)+128 a^4 C+3 b^4 (49 A+37 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)}}{9 b d}","\frac{2 \left(48 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{4 a \left(32 a^2 C+42 A b^2+31 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^4 d}-\frac{2 a \left(4 a^2 b^2 (42 A+19 C)+128 a^4 C+3 b^4 (49 A+37 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)}}{9 b d}",1,"(2*(128*a^4*C + 21*b^4*(9*A + 7*C) + 12*a^2*b^2*(14*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(128*a^4*C + 4*a^2*b^2*(42*A + 19*C) + 3*b^4*(49*A + 37*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(42*A*b^2 + 32*a^2*C + 31*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^4*d) + (2*(48*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) - (16*a*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b*d)","A",9,8,35,0.2286,1,"{3050, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
649,1,305,0,0.5891958,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(24 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}+\frac{2 \left(2 a^2 b^2 (35 A+16 C)+48 a^4 C+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(24 a^2 C+35 A b^2+22 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{12 a C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}","\frac{2 \left(24 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}+\frac{2 \left(2 a^2 b^2 (35 A+16 C)+48 a^4 C+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(24 a^2 C+35 A b^2+22 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{12 a C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}",1,"(-4*a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(48*a^4*C + 5*b^4*(7*A + 5*C) + 2*a^2*b^2*(35*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) - (12*a*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)","A",8,8,35,0.2286,1,"{3050, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
650,1,233,0,0.3483871,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 a \left(8 a^2 C+15 A b^2+7 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}","-\frac{2 a \left(8 a^2 C+15 A b^2+7 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*(8*a^2*C + 3*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(15*A*b^2 + 8*a^2*C + 7*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)","A",7,7,33,0.2121,1,"{3050, 3023, 2752, 2663, 2661, 2655, 2653}"
651,1,174,0,0.2057932,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2 C+b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","\frac{2 \left(2 a^2 C+b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-4*a*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*C + b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,27,0.2222,1,"{3024, 2752, 2663, 2661, 2655, 2653}"
652,1,183,0,0.4112636,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,33,0.2424,1,"{3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
653,1,214,0,0.6303068,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}","\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"-((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)","A",9,9,35,0.2571,1,"{3056, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
654,1,278,0,0.884024,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"(3*A*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",10,10,35,0.2857,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
655,1,370,0,1.3192609,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}+\frac{\left(8 a^2 (2 A+3 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(4 a^2 (A+2 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}+\frac{\left(8 a^2 (2 A+3 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(4 a^2 (A+2 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"-((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((5*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^3*d) - (5*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",11,10,35,0.2857,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
656,1,473,0,1.129866,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+7 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(48 a^2 C+35 A b^2-13 b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^4 d \left(a^2-b^2\right)}+\frac{2 \left(4 a^2 b^2 (70 A+29 C)+384 a^4 C+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+7 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(48 a^2 C+35 A b^2-13 b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^4 d \left(a^2-b^2\right)}+\frac{2 \left(4 a^2 b^2 (70 A+29 C)+384 a^4 C+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(384*a^4*C + 5*b^4*(7*A + 5*C) + 4*a^2*b^2*(70*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*a^2*b^2*(70*A - 31*C) + 192*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^4*(a^2 - b^2)*d) - (2*a*(35*A*b^2 + 48*a^2*C - 13*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)","A",9,8,35,0.2286,1,"{3048, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
657,1,375,0,0.7351576,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 C+5 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2 C+5 A b^2-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{4 a \left(2 C \left(4 a^2+b^2\right)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 C+5 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2 C+5 A b^2-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{4 a \left(2 C \left(4 a^2+b^2\right)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b^2 (5 A-4 C)+16 a^4 C-b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(5*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(5*A*b^2 + 2*(4*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(5*A*b^2 + 8*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 + 6*a^2*C - b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",8,8,35,0.2286,1,"{3048, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
658,1,256,0,0.423847,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(C \left(8 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2 C+3 A b^2-5 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}","\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(C \left(8 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2 C+3 A b^2-5 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(-2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 + (8*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)","A",7,7,33,0.2121,1,"{3032, 3023, 2752, 2663, 2661, 2655, 2653}"
659,1,202,0,0.246881,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,27,0.2222,1,"{3022, 2752, 2663, 2661, 2655, 2653}"
660,1,259,0,0.7020167,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(A*b^2 + a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",9,9,33,0.2727,1,"{3056, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
661,1,296,0,0.967668,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{b \left(3 A b^2-a^2 (A-2 C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-a^2 (A-2 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}","-\frac{b \left(3 A b^2-a^2 (A-2 C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-a^2 (A-2 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"((3*A*b^2 - a^2*(A - 2*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - a^2*(A - 2*C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,35,0.2857,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
662,1,370,0,1.3409237,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b^2 \left(15 A b^2-a^2 (7 A-8 C)\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \left(15 A b^2-a^2 (7 A-8 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}","\frac{b^2 \left(15 A b^2-a^2 (7 A-8 C)\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \left(15 A b^2-a^2 (7 A-8 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}",1,"-(b*(15*A*b^2 - a^2*(7*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (5*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b^2*(15*A*b^2 - a^2*(7*A - 8*C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])","A",11,10,35,0.2857,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
663,1,521,0,1.3356924,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 \left(-a^2 b^2 (A-6 C)-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 b^2 (15 A-71 C)+48 a^4 C-b^4 (35 A-3 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}-\frac{4 a \left(a^2 b^2 (10 A-49 C)+32 a^4 C-b^4 (20 A-7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(4 a^2 b^2 (10 A-29 C)+128 a^4 C-b^4 (45 A+17 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^4 b^2 (10 A-53 C)-5 a^2 b^4 (15 A-11 C)+128 a^6 C+3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 \left(-a^2 b^2 (A-6 C)-4 a^4 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 b^2 (15 A-71 C)+48 a^4 C-b^4 (35 A-3 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}-\frac{4 a \left(a^2 b^2 (10 A-49 C)+32 a^4 C-b^4 (20 A-7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(4 a^2 b^2 (10 A-29 C)+128 a^4 C-b^4 (45 A+17 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^4 b^2 (10 A-53 C)-5 a^2 b^4 (15 A-11 C)+128 a^6 C+3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(4*a^4*b^2*(10*A - 53*C) - 5*a^2*b^4*(15*A - 11*C) + 128*a^6*C + 3*b^6*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(4*a^2*b^2*(10*A - 29*C) + 128*a^4*C - b^4*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*(3*A*b^4 - a^2*b^2*(A - 6*C) - 4*a^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(a^2*b^2*(10*A - 49*C) - b^4*(20*A - 7*C) + 32*a^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) + (2*(a^2*b^2*(15*A - 71*C) - b^4*(35*A - 3*C) + 48*a^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)","A",9,9,35,0.2571,1,"{3048, 3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
664,1,392,0,0.8296362,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{4 a \left(5 a^2 b^2 C-3 a^4 C+2 A b^4\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2 b^2 (A-14 C)+8 a^4 C-b^4 (3 A-4 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{4 a \left(5 a^2 b^2 C-3 a^4 C+2 A b^4\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2 b^2 (A-14 C)+8 a^4 C-b^4 (3 A-4 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-4*a*(a^2*b^2*(A - 14*C) - b^4*(3*A - 4*C) + 8*a^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*b^2*(A - 8*C) + 16*a^4*C - b^4*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",8,8,35,0.2286,1,"{3048, 3031, 3023, 2752, 2663, 2661, 2655, 2653}"
665,1,314,0,0.5096114,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(-8 a^2 C+A b^2+9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b^2 (A+15 C)-8 a^4 C+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(-8 a^2 C+A b^2+9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b^2 (A+15 C)-8 a^4 C+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*a*(A*b^2 - 8*a^2*C + 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,33,0.2121,1,"{3032, 3021, 2752, 2663, 2661, 2655, 2653}"
666,1,298,0,0.3900845,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{4 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(2 A b^2-C \left(a^2-3 b^2\right)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{4 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(2 A b^2-C \left(a^2-3 b^2\right)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(4*a*(2*A*b^2 - (a^2 - 3*b^2)*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,27,0.2593,1,"{3022, 2754, 2752, 2663, 2661, 2655, 2653}"
667,1,375,0,1.0811703,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}",1,"(2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 + a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,33,0.3030,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
668,1,416,0,1.4047176,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{b \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \left(5 A b^2-a^2 (3 A-2 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(5 A b^2-a^2 (3 A-2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}","-\frac{b \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \left(5 A b^2-a^2 (3 A-2 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(5 A b^2-a^2 (3 A-2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"((26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b^2 - a^2*(3*A - 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 - a^2*(3*A - 2*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))","A",11,10,35,0.2857,1,"{3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
669,1,389,0,0.6053871,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2 (-C)+4 A b^2+5 b^2 C\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left(4 A b^2-C \left(a^2-5 b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \left(-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2 (-C)+4 A b^2+5 b^2 C\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left(4 A b^2-C \left(a^2-5 b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-a^2 b^2 (23 A+19 C)+2 a^4 C-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(4*A*b^2 - (a^2 - 5*b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) - (4*a*(4*A*b^2 - a^2*C + 5*b^2*C)*Sin[c + d*x])/(15*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,27,0.2593,1,"{3022, 2754, 2752, 2663, 2661, 2655, 2653}"
670,1,157,0,0.2333762,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,7,32,0.2188,1,"{3016, 2753, 2752, 2663, 2661, 2655, 2653}"
671,1,116,0,0.157303,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,32,0.1875,1,"{3016, 2752, 2663, 2661, 2655, 2653}"
672,1,165,0,0.2390533,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{4 a b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","-\frac{4 a b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,32,0.2188,1,"{3016, 2754, 2752, 2663, 2661, 2655, 2653}"
673,1,243,0,0.3411804,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Int[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","-\frac{2 b \left(5 a^2+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(5 a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 b \left(5 a^2+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(5 a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(5*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(5*a^2 + 3*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,32,0.2188,1,"{3016, 2754, 2752, 2663, 2661, 2655, 2653}"
674,1,196,0,0.237284,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{10 b (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 b (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}","\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{10 b (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 b (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}",1,"(2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*b*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*b*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*b*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)","A",8,6,33,0.1818,1,"{3034, 3023, 2748, 2635, 2639, 2641}"
675,1,165,0,0.2071687,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",7,6,33,0.1818,1,"{3034, 3023, 2748, 2635, 2641, 2639}"
676,1,134,0,0.1854413,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,6,33,0.1818,1,"{3034, 3023, 2748, 2639, 2635, 2641}"
677,1,101,0,0.1729789,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*b*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,33,0.1515,1,"{3034, 3023, 2748, 2641, 2639}"
678,1,95,0,0.173419,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*b*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,33,0.1515,1,"{3032, 3023, 2748, 2641, 2639}"
679,1,95,0,0.1856463,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b (A-C) 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+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b (A-C) 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*b*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,33,0.1515,1,"{3032, 3021, 2748, 2641, 2639}"
680,1,132,0,0.1951108,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*A*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{3032, 3021, 2748, 2636, 2639, 2641}"
681,1,165,0,0.2146788,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (3 A+5 C) \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 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (3 A+5 C) \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*b*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*A*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,33,0.1818,1,"{3032, 3021, 2748, 2636, 2641, 2639}"
682,1,254,0,0.4795925,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 a b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}","\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 a b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(4*a*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (8*a*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)","A",8,7,35,0.2000,1,"{3050, 3033, 3023, 2748, 2635, 2641, 2639}"
683,1,205,0,0.4242982,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}","\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(2*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a*b*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (8*a*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)","A",7,7,35,0.2000,1,"{3050, 3033, 3023, 2748, 2639, 2635, 2641}"
684,1,171,0,0.4086971,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{4 a b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}","\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{4 a b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(4*a*b*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(4*a^2*C + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (8*a*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)","A",6,6,35,0.1714,1,"{3050, 3033, 3023, 2748, 2641, 2639}"
685,1,166,0,0.4061326,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (3 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (3 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(-2*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a*b*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a*b*(3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,35,0.1714,1,"{3048, 3033, 3023, 2748, 2641, 2639}"
686,1,154,0,0.4002104,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-4*a*b*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (8*a*A*b*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,35,0.1714,1,"{3048, 3031, 3023, 2748, 2641, 2639}"
687,1,169,0,0.4082094,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a*b*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (8*a*A*b*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(4*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,6,35,0.1714,1,"{3048, 3031, 3021, 2748, 2641, 2639}"
688,1,203,0,0.4486898,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a*b*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a*A*b*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(4*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,7,35,0.2000,1,"{3048, 3031, 3021, 2748, 2636, 2639, 2641}"
689,1,295,0,0.7595144,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d}+\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{165 d}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}+\frac{4 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{33 d}","\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d}+\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{165 d}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}+\frac{4 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{33 d}",1,"(2*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(165*d) + (2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(231*d) + (4*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(33*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)","A",8,8,35,0.2286,1,"{3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
690,1,245,0,0.7024881,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{4 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}","\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{4 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}",1,"(2*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(63*A*b^2 + 8*a^2*C + 45*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(24*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (4*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)","A",7,7,35,0.2000,1,"{3050, 3049, 3033, 3023, 2748, 2641, 2639}"
691,1,244,0,0.7379129,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}","\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*a*b^2*(35*A - 11*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,35,0.2000,1,"{3048, 3049, 3033, 3023, 2748, 2641, 2639}"
692,1,218,0,0.6909809,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 a b^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^3 (35 A-3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 a b^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^3 (35 A-3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}",1,"(-2*b*(15*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a*b^2*(5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d - (2*b^3*(35*A - 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,7,35,0.2000,1,"{3048, 3047, 3033, 3023, 2748, 2641, 2639}"
693,1,229,0,0.6794846,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^3 (9 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}","\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^3 (9 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}",1,"(-2*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(8*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (2*b^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,35,0.2000,1,"{3048, 3047, 3031, 3023, 2748, 2641, 2639}"
694,1,243,0,0.7246995,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (6*b*(8*A*b^2 + 7*a^2*(3*A + 5*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (12*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,7,35,0.2000,1,"{3048, 3047, 3031, 3021, 2748, 2641, 2639}"
695,1,293,0,0.7882608,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,8,35,0.2286,1,"{3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
696,1,382,0,1.1537992,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{2 \left(11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6435 d}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac{16 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}","\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{2 \left(11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6435 d}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac{16 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}",1,"(2*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/(195*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(48*a^2*C + 11*b^2*(13*A + 11*C))*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (16*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)","A",9,8,35,0.2286,1,"{3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
697,1,329,0,1.0862096,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{231 d}+\frac{2 \left(9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}+\frac{16 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}","\frac{2 \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{231 d}+\frac{2 \left(9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}+\frac{16 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}",1,"(8*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (4*a*b*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(16*a^2*C + 3*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (16*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)","A",8,7,35,0.2000,1,"{3050, 3049, 3033, 3023, 2748, 2641, 2639}"
698,1,320,0,1.1789828,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}-\frac{2 a b (21 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}","\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}-\frac{2 a b (21 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}",1,"(-2*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a*b*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a*b*(a^2*(63*A - 31*C) - 6*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) - (2*b^2*(3*a^2*(105*A - 41*C) - 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) - (2*a*b*(21*A - 5*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,7,35,0.2000,1,"{3048, 3049, 3033, 3023, 2748, 2641, 2639}"
699,1,300,0,1.1487553,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 b^2 (21 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}","\frac{2 \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 b^2 (21 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}",1,"(-8*a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b^2*(3*a^2*(49*A - 13*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (4*a*b^3*(175*A - 27*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b^2*(21*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,8,35,0.2286,1,"{3048, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
700,1,321,0,1.2248579,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)-b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)-b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a*b*(2*b^2*(33*A - 5*C) + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(16*A*b^2 + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,7,35,0.2000,1,"{3048, 3047, 3033, 3023, 2748, 2641, 2639}"
701,1,316,0,1.1431595,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-8*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a*b*(96*A*b^2 + a^2*(101*A + 175*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,7,35,0.2000,1,"{3048, 3047, 3031, 3023, 2748, 2641, 2639}"
702,1,325,0,1.1557174,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 b^4 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+192 A b^4\right) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 b^4 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+192 A b^4\right) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a*b*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a*b*(32*A*b^2 + a^2*(101*A + 147*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(192*A*b^4 + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(48*A*b^2 + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,7,35,0.2000,1,"{3048, 3047, 3031, 3021, 2748, 2641, 2639}"
703,1,377,0,1.23302,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",9,8,35,0.2286,1,"{3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
704,1,299,0,1.5017081,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 a \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d}+\frac{2 \left(3 a^2 b^2 (5 A+3 C)+15 a^4 C+b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 b^5 d}+\frac{2 a^4 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d (a+b)}+\frac{2 \left(9 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 b^3 d}-\frac{2 a \left(7 a^2 C+7 A b^2+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^4 d}-\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 b d}","-\frac{2 a \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d}+\frac{2 \left(3 a^2 b^2 (5 A+3 C)+15 a^4 C+b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 b^5 d}+\frac{2 a^4 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d (a+b)}+\frac{2 \left(9 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 b^3 d}-\frac{2 a \left(7 a^2 C+7 A b^2+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^4 d}-\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 b d}",1,"(2*(15*a^4*C + 3*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*b^5*d) - (2*a*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*b^6*d) + (2*a^4*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^6*(a + b)*d) - (2*a*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^4*d) + (2*(9*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*b^3*d) - (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b^2*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*b*d)","A",9,7,35,0.2000,1,"{3050, 3049, 3059, 2639, 3002, 2641, 2805}"
705,1,239,0,1.120372,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}-\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}","\frac{2 \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}-\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"(-2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*b^4*d) + (2*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^5*(a + b)*d) + (2*(7*a^2*C + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) - (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,7,35,0.2000,1,"{3050, 3049, 3059, 2639, 3002, 2641, 2805}"
706,1,181,0,0.7888702,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 a \left(C \left(3 a^2+b^2\right)+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}-\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}","-\frac{2 a \left(C \left(3 a^2+b^2\right)+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}-\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"(2*(5*a^2*C + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) - (2*a*(3*A*b^2 + (3*a^2 + b^2)*C)*EllipticF[(c + d*x)/2, 2])/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^4*(a + b)*d) - (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,35,0.2000,1,"{3050, 3049, 3059, 2639, 3002, 2641, 2805}"
707,1,130,0,0.5340397,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(-2*a*C*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*(3*a^2*C + b^2*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,35,0.1714,1,"{3050, 3059, 2639, 3002, 2641, 2805}"
708,1,85,0,0.2719095,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*C*EllipticF[(c + d*x)/2, 2])/(b^2*d) + (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",5,5,35,0.1429,1,"{3060, 2639, 3002, 2641, 2805}"
709,1,112,0,0.5014976,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","-\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*A*EllipticE[(c + d*x)/2, 2])/(a*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(b*d) - (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*b*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])","A",6,6,35,0.1714,1,"{3056, 3059, 2639, 3002, 2641, 2805}"
710,1,140,0,0.7295764,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}+\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 A b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}+\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 A b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*A*b*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*A*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*A*b*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])","A",7,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
711,1,206,0,1.0732603,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 A b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 A b \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 A b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 A b \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(5*A*b^2 + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*A*b*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*A*b*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])","A",8,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
712,1,270,0,1.4943641,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d}+\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d}+\frac{2 b^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}+\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d}+\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d}+\frac{2 b^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}+\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*b*(5*A*b^2 + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^4*d) + (2*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^3*d) + (2*b^2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*A*b*Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Cos[c + d*x]^(3/2)) - (2*b*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c + d*x]])","A",9,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
713,1,344,0,1.9356974,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(11/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d}-\frac{2 \left(3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^5 d}-\frac{2 b^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^5 d (a+b)}-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 (7 A+9 C)+9 A b^2\right) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}","-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d}-\frac{2 \left(3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^5 d}-\frac{2 b^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^5 d (a+b)}-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 (7 A+9 C)+9 A b^2\right) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*a^5*d) - (2*b*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^4*d) - (2*b^3*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^5*(a + b)*d) + (2*A*Sin[c + d*x])/(9*a*d*Cos[c + d*x]^(9/2)) - (2*A*b*Sin[c + d*x])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*(9*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*a^3*d*Cos[c + d*x]^(5/2)) - (2*b*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^4*d*Cos[c + d*x]^(3/2)) + (2*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sin[c + d*x])/(15*a^5*d*Sqrt[Cos[c + d*x]])","A",10,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
714,1,370,0,1.3758279,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{a \left(a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}","-\frac{a \left(a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}",1,"((3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*b^4*(a^2 - b^2)*d) - (a*(a^2*b^2*(9*A - 20*C) + 21*a^4*C - 4*b^4*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^5*(a + b)^2*d) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{3048, 3049, 3059, 2639, 3002, 2641, 2805}"
715,1,292,0,0.9694795,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 b^2 (3 A-16 C)+15 a^4 C-2 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\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(-a^2 b^2 (A-7 C)-5 a^4 C+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}","\frac{\left(a^2 b^2 (3 A-16 C)+15 a^4 C-2 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\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(-a^2 b^2 (A-7 C)-5 a^4 C+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}",1,"-((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d)) + ((a^2*b^2*(3*A - 16*C) + 15*a^4*C - 2*b^4*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{3048, 3049, 3059, 2639, 3002, 2641, 2805}"
716,1,217,0,0.6622859,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\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 b^2 (A+5 C)-3 a^4 C+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\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 b^2 (A+5 C)-3 a^4 C+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((A*b^2 + 3*a^2*C - 2*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((A*b^4 - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{3048, 3059, 2639, 3002, 2641, 2805}"
717,1,214,0,0.7053477,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^4 C+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^4 C+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-(((A*b^2 + a^2*C)*EllipticE[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{3056, 3059, 2639, 3002, 2641, 2805}"
718,1,270,0,1.0657891,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\left(a^2 C+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}","\frac{\left(a^2 C+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}",1,"((3*A*b^2 - a^2*(2*A - C))*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*EllipticF[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a^2*(2*A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
719,1,336,0,1.4532715,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (7 A-C)-3 a^4 C+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (7 A-C)-3 a^4 C+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"-((b*(5*A*b^2 - a^2*(4*A - C))*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - a^2*(2*A - 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + (b*(5*A*b^2 - a^2*(4*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
720,1,427,0,1.8072596,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2),x]","\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \left(a^2-b^2\right)}+\frac{\left(-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+35 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d \left(a^2-b^2\right)}+\frac{b \left(-3 a^2 b^2 (3 A-C)-5 a^4 C+7 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(7 A b^2-a^2 (2 A-5 C)\right) \sin (c+d x)}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}-\frac{\left(-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+35 A b^4\right) \sin (c+d x)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \left(a^2-b^2\right)}+\frac{\left(-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+35 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d \left(a^2-b^2\right)}+\frac{b \left(-3 a^2 b^2 (3 A-C)-5 a^4 C+7 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(7 A b^2-a^2 (2 A-5 C)\right) \sin (c+d x)}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}-\frac{\left(-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+35 A b^4\right) \sin (c+d x)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"((35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^4*(a^2 - b^2)*d) + (b*(7*A*b^2 - a^2*(4*A - 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^3*(a^2 - b^2)*d) + (b*(7*A*b^4 - 3*a^2*b^2*(3*A - C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^4*(a - b)*(a + b)^2*d) - ((7*A*b^2 - a^2*(2*A - 5*C))*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (b*(7*A*b^2 - a^2*(4*A - 3*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x]))","A",9,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
721,1,433,0,1.601151,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(a^2 b^2 (3 A-65 C)+35 a^4 C-3 b^4 (3 A-8 C)\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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (A+13 C)-7 a^4 C+5 A b^4\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (3 A-61 C)+35 a^4 C-b^4 (21 A-8 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}","\frac{\left(a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(a^2 b^2 (3 A-65 C)+35 a^4 C-3 b^4 (3 A-8 C)\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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (A+13 C)-7 a^4 C+5 A b^4\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (3 A-61 C)+35 a^4 C-b^4 (21 A-8 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}",1,"-(a*(a^2*b^2*(3*A - 65*C) - 3*b^4*(3*A - 8*C) + 35*a^4*C)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) + ((a^4*b^2*(9*A - 223*C) - a^2*b^4*(15*A - 128*C) + 105*a^6*C + 8*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((a^2*b^2*(3*A - 61*C) - b^4*(21*A - 8*C) + 35*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((5*A*b^4 - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,35,0.2286,1,"{3048, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
722,1,345,0,1.1407535,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(-a^2 b^2 (A+33 C)+15 a^4 C+b^4 (7 A+24 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^2 (A+29 C)-15 a^4 C+b^4 (5 A-8 C)\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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+15 a^6 C+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A+11 C)-5 a^4 C+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}","-\frac{a \left(-a^2 b^2 (A+33 C)+15 a^4 C+b^4 (7 A+24 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^2 (A+29 C)-15 a^4 C+b^4 (5 A-8 C)\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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+15 a^6 C+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A+11 C)-5 a^4 C+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"-((b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*C + b^4*(7*A + 24*C) - a^2*b^2*(A + 33*C))*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{3048, 3047, 3059, 2639, 3002, 2641, 2805}"
723,1,348,0,1.1299993,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)-3 a^6 C+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)-3 a^6 C+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}-\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{3048, 3055, 3059, 2639, 3002, 2641, 2805}"
724,1,345,0,1.0733383,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\left(-7 a^2 b^2 (A+C)+a^4 C+A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+a^6 (-C)+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}-\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(-7 a^2 b^2 (A+C)+a^4 C+A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+a^6 (-C)+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}-\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
725,1,417,0,1.5474126,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+3 a^6 C+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}","-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+3 a^6 C+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}",1,"-((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*EllipticF[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
726,1,494,0,2.0959617,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\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{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}-\frac{\left(-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}","\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\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{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}-\frac{\left(-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"(b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*EllipticF[(c + d*x)/2, 2])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",9,7,35,0.2000,1,"{3056, 3055, 3059, 2639, 3002, 2641, 2805}"
727,1,553,0,1.5317207,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}",1,"((a - b)*Sqrt[a + b]*(3*a^2*C - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) - (Sqrt[a + b]*(3*a^2*C - 2*a*b*C - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) - (a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) - ((3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) - (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)","A",8,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
728,1,455,0,1.0832585,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}",1,"-((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (Sqrt[a + b]*(8*A*b + (a + 2*b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (Sqrt[a + b]*(a^2*C - 4*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,37,0.1892,1,"{3050, 3061, 3053, 2809, 2998, 2816, 2994}"
729,1,439,0,1.0612702,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"((a - b)*Sqrt[a + b]*(2*A - C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a*A - 2*A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,37,0.1892,1,"{3048, 3061, 3053, 2809, 2998, 2816, 2994}"
730,1,394,0,0.7687621,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*A*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) - (2*Sqrt[a + b]*(A*b - a*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,37,0.1622,1,"{3048, 3053, 2809, 2998, 2816, 2994}"
731,1,345,0,0.8401757,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b + 15*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{3048, 3055, 2998, 2816, 2994}"
732,1,415,0,1.1885859,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*b*Sqrt[a + b]*(8*A*b^2 + a^2*(19*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b + 8*A*b^2 + 5*a^2*(5*A + 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^2 - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))","A",6,5,37,0.1351,1,"{3048, 3055, 2998, 2816, 2994}"
733,1,638,0,1.9947585,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{64 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 b C+3 a^3 C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}-\frac{\sqrt{a+b} \left(24 a^2 b^2 (2 A+C)+3 a^4 C+16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{8 b d}","-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{64 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 b C+3 a^3 C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}-\frac{\sqrt{a+b} \left(24 a^2 b^2 (2 A+C)+3 a^4 C+16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{8 b d}",1,"-((a - b)*Sqrt[a + b]*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) - (Sqrt[a + b]*(3*a^3*C - 2*a^2*b*C - 8*b^3*(4*A + 3*C) - 4*a*b^2*(20*A + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) - (Sqrt[a + b]*(3*a^4*C + 24*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(64*b^2*d*Sqrt[Cos[c + d*x]]) - ((3*a^2*C - 4*b^2*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) - (a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)","A",9,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
734,1,553,0,1.5807951,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}",1,"-((a - b)*Sqrt[a + b]*(3*a^2*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d) + (Sqrt[a + b]*(48*a*A*b + 24*A*b^2 + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (a*Sqrt[a + b]*(24*A*b^2 - a^2*C + 12*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
735,1,509,0,1.5532313,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(8*A - 5*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*a*A - 16*A*b - 5*a*C - 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*A*b^2 + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (a*(8*A - 5*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,8,37,0.2162,1,"{3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
736,1,500,0,1.525253,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((a - b)*b*Sqrt[a + b]*(8*A - 3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A + 3*C) - a*(8*A*b - 3*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (3*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,8,37,0.2162,1,"{3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
737,1,465,0,1.1759125,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*(A*b^2 + a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d) - (2*Sqrt[a + b]*(A*b^2 - 2*a*b*(2*A + 5*C) + a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,37,0.1892,1,"{3048, 3047, 3053, 2809, 2998, 2816, 2994}"
738,1,418,0,1.2319908,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*(a - b)*b*Sqrt[a + b]*(3*A*b^2 - a^2*(41*A + 70*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*(a - b)*Sqrt[a + b]*(25*a^2*A - 57*a*A*b - 6*A*b^2 + 35*a^2*C - 105*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (6*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(3*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",6,6,37,0.1622,1,"{3048, 3047, 3055, 2998, 2816, 2994}"
739,1,502,0,1.7362092,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (39 A b+63 b C)-21 a^3 (7 A+9 C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (39 A b+63 b C)-21 a^3 (7 A+9 C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^2 + 8*A*b^3 - 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(3*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) - (4*b*(2*A*b^2 - a^2*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",7,6,37,0.1622,1,"{3048, 3047, 3055, 2998, 2816, 2994}"
740,1,746,0,2.7577754,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{a \left(-15 a^2 C+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}-\frac{\left(-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-12 a^2 b^2 (220 A+141 C)-30 a^3 b C+45 a^4 C-8 a b^3 (260 A+193 C)-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}+\frac{(a-b) \sqrt{a+b} \left(-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}-\frac{a \sqrt{a+b} \left(40 a^2 b^2 (2 A+C)+3 a^4 C+80 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}-\frac{3 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}","-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{a \left(-15 a^2 C+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}-\frac{\left(-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-12 a^2 b^2 (220 A+141 C)-30 a^3 b C+45 a^4 C-8 a b^3 (260 A+193 C)-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}+\frac{(a-b) \sqrt{a+b} \left(-12 a^2 b^2 (220 A+141 C)+45 a^4 C-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}-\frac{a \sqrt{a+b} \left(40 a^2 b^2 (2 A+C)+3 a^4 C+80 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}-\frac{3 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}",1,"((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) - ((45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) - (3*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)","A",10,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
741,1,635,0,2.0579263,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (192 A+59 C)+15 a^3 C+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}+\frac{\sqrt{a+b} \left(-120 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}","\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (192 A+59 C)+15 a^3 C+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}+\frac{\sqrt{a+b} \left(-120 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}",1,"-((a - b)*Sqrt[a + b]*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) + (Sqrt[a + b]*(15*a^3*C + 24*b^3*(4*A + 3*C) + 2*a^2*b*(192*A + 59*C) + 4*a*b^2*(108*A + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) + (Sqrt[a + b]*(5*a^4*C - 120*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",9,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
742,1,609,0,2.1136491,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) - (Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C) - 2*a*b*(72*A + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (5*a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) - ((a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) - (a*b*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",9,8,37,0.2162,1,"{3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
743,1,567,0,1.9814563,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}",1,"((a - b)*b*Sqrt[a + b]*(56*A - 27*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) + (Sqrt[a + b]*(6*b^2*(12*A + C) + 8*a^2*(A + 3*C) - a*(56*A*b - 27*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (a*b*(56*A - 27*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b^2*(8*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",9,9,37,0.2432,1,"{3048, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
744,1,606,0,2.0305517,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (34 A b+90 b C)-6 a^3 (3 A+5 C)-a b^2 (46 A-15 C)+30 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{5 a b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (34 A b+90 b C)-6 a^3 (3 A+5 C)-a b^2 (46 A-15 C)+30 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{5 a b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((a - b)*Sqrt[a + b]*(b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (Sqrt[a + b]*(30*A*b^3 - a*b^2*(46*A - 15*C) - 6*a^3*(3*A + 5*C) + a^2*(34*A*b + 90*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (5*a*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - ((b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",9,8,37,0.2162,1,"{3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
745,1,540,0,1.5616876,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \left(a^2 b (29 A+49 C)+a^3 (-(5 A+7 C))-9 a b^2 (3 A+7 C)+3 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \left(a^2 b (29 A+49 C)+a^3 (-(5 A+7 C))-9 a b^2 (3 A+7 C)+3 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*b*Sqrt[a + b]*(3*A*b^2 + a^2*(29*A + 49*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d) - (2*Sqrt[a + b]*(3*A*b^3 - 9*a*b^2*(3*A + 7*C) - a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(3*A*b^2 + a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,7,37,0.1892,1,"{3048, 3047, 3053, 2809, 2998, 2816, 2994}"
746,1,504,0,1.7580852,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 A+28 C)+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)+10 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 A+28 C)+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)+10 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 21*a^3*(7*A + 9*C) + 15*a*b^2*(11*A + 21*C) - 6*a^2*b*(19*A + 28*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*(15*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(5*A*b^2 + a^2*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",7,6,37,0.1622,1,"{3048, 3047, 3055, 2998, 2816, 2994}"
747,1,587,0,2.5454008,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","-\frac{2 \left(-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}","-\frac{2 \left(-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*(5*A*b^2 + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",8,6,37,0.1622,1,"{3048, 3047, 3055, 2998, 2816, 2994}"
748,1,554,0,1.5234072,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(15 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^2 C-10 a b C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}+\frac{a \sqrt{a+b} \left(5 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}-\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","\frac{\left(15 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^2 C-10 a b C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}+\frac{a \sqrt{a+b} \left(5 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}-\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"-((a - b)*Sqrt[a + b]*(15*a^2*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d) + (Sqrt[a + b]*(15*a^2*C - 10*a*b*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) + (a*Sqrt[a + b]*(8*A*b^2 + 5*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) + ((15*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) - (5*a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",8,8,37,0.2162,1,"{3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
749,1,455,0,1.0546414,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}-\frac{3 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{C (3 a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{3 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}-\frac{3 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{C (3 a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{3 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"(3*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - ((3*a - 2*b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - (Sqrt[a + b]*(3*a^2*C + 4*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) - (3*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",7,7,37,0.1892,1,"{3050, 3061, 3053, 2809, 2998, 2816, 2994}"
750,1,393,0,0.7279061,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}+\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}+\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"-(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*(2*A*b + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d) + (a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",6,6,37,0.1622,1,"{3062, 3053, 2809, 2998, 2816, 2994}"
751,1,343,0,0.4446286,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)","A",6,6,37,0.1622,1,"{3054, 2809, 12, 2801, 2816, 2994}"
752,1,283,0,0.5082209,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{4 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{4 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*A*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(2*A*b + a*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))","A",4,4,37,0.1081,1,"{3056, 2998, 2816, 2994}"
753,1,354,0,0.8101061,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{8 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{8 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) + (2*Sqrt[a + b]*(2*a*A*b - 8*A*b^2 - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (8*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
754,1,429,0,1.1977834,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \left(-a^2 (44 A b+70 b C)-5 a^3 (5 A+7 C)+12 a A b^2-48 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}-\frac{12 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \left(-a^2 (44 A b+70 b C)-5 a^3 (5 A+7 C)+12 a A b^2-48 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}-\frac{12 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*(a - b)*b*Sqrt[a + b]*(24*A*b^2 + a^2*(22*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d) - (2*Sqrt[a + b]*(12*a*A*b^2 - 48*A*b^3 - 5*a^3*(5*A + 7*C) - a^2*(44*A*b + 70*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (12*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Cos[c + d*x]^(3/2))","A",6,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
755,1,604,0,1.7142187,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}",1,"((8*A*b^2 + 15*a^2*C - 7*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - ((8*A*b^2 + (15*a^2 + 5*a*b - 2*b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((4*A*b^2 + 5*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)","A",8,8,37,0.2162,1,"{3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
756,1,503,0,1.2381526,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"-(((2*A*b^2 + 3*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d)) + ((2*A*b^2 + a*(3*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) + (3*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",7,7,37,0.1892,1,"{3048, 3061, 3053, 2809, 2998, 2816, 2994}"
757,1,421,0,0.8022601,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 (A b-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 (A b-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}",1,"(2*(A*b^2 + a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d) + (2*(A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{3052, 2809, 2993, 2998, 2816, 2994}"
758,1,308,0,0.6245561,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 (a (A-C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 (a (A-C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}",1,"(-2*(2*A*b^2 - a^2*(A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(2*A*b + a*(A - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,37,0.1081,1,"{3056, 2998, 2816, 2994}"
759,1,392,0,0.979306,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}",1,"(2*b*(8*A*b^2 - a^2*(5*A - 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d) + (2*(6*a*A*b + 8*A*b^2 + a^2*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
760,1,494,0,1.4460827,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^2 b (2 A+5 C)+a^3 (3 A+5 C)+12 a A b^2+16 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}-\frac{2 \left(-2 a^2 b^2 (4 A-5 C)+a^4 (-(3 A+5 C))+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}","\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^2 b (2 A+5 C)+a^3 (3 A+5 C)+12 a A b^2+16 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}-\frac{2 \left(-2 a^2 b^2 (4 A-5 C)+a^4 (-(3 A+5 C))+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}",1,"(-2*(16*A*b^4 - 2*a^2*b^2*(4*A - 5*C) - a^4*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d) - (2*(12*a*A*b^2 + 16*A*b^3 + 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (2*b*(8*A*b^2 - a^2*(3*A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",6,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
761,1,650,0,2.4723539,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(-21 a^2 b^2 C+5 a^3 b C+15 a^4 C+2 a A b^3-3 a b^3 C-6 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{\left(26 a^2 b^2 C-15 a^4 C+8 A b^4-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{5 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(-21 a^2 b^2 C+5 a^3 b C+15 a^4 C+2 a A b^3-3 a b^3 C-6 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{\left(26 a^2 b^2 C-15 a^4 C+8 A b^4-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{5 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) + ((2*a*A*b^3 - 6*A*b^4 + 15*a^4*C + 5*a^3*b*C - 21*a^2*b^2*C - 3*a*b^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) + (5*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])","A",8,8,37,0.2162,1,"{3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
762,1,563,0,1.6133744,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b C-3 a^3 C+3 a A b^2+6 a b^2 C-A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b C-3 a^3 C+3 a A b^2+6 a b^2 C-A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(-2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(3*a*A*b^2 - A*b^3 - 3*a^3*C - a^2*b*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{3048, 3051, 2809, 2993, 2998, 2816, 2994}"
763,1,417,0,1.0299197,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 (-(3 A+C))+3 a b (A+C)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{4 b \left(3 a^2 A+2 a^2 C-A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 (-(3 A+C))+3 a b (A+C)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{4 b \left(3 a^2 A+2 a^2 C-A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*A*b^2 + 3*a*b*(A + C) - a^2*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*b*(A*b^2 - a^2*(3*A + 2*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,37,0.1351,1,"{3056, 2993, 2998, 2816, 2994}"
764,1,449,0,1.1575926,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","-\frac{4 \left(-a^2 b^2 (4 A+C)+a^4 (-C)+2 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b (9 A+C)-3 a^3 (A-C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}","-\frac{4 \left(-a^2 b^2 (4 A+C)+a^4 (-C)+2 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b (9 A+C)-3 a^3 (A-C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(2*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(6*a*A*b^2 + 8*A*b^3 - 3*a^3*(A - C) - a^2*b*(9*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) - (4*(2*A*b^4 - a^4*C - a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
765,1,549,0,1.6722925,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 \left(5 a^2 A b^2+2 a^4 C-3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 b^2 (8 A-C)-a^3 (9 A b-3 b C)+a^4 (-(A+3 C))+12 a A b^3+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{4 b \left(-a^2 b^2 (14 A-C)+a^4 (4 A-3 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}","\frac{2 \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 \left(5 a^2 A b^2+2 a^4 C-3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 b^2 (8 A-C)-a^3 (9 A b-3 b C)+a^4 (-(A+3 C))+12 a A b^3+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{4 b \left(-a^2 b^2 (14 A-C)+a^4 (4 A-3 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(-4*b*(8*A*b^4 + a^4*(4*A - 3*C) - a^2*b^2*(14*A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(12*a*A*b^3 + 16*A*b^4 - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C) - a^3*(9*A*b - 3*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))","A",6,5,37,0.1351,1,"{3056, 3055, 2998, 2816, 2994}"
766,1,318,0,0.8584772,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+b^2 (m+1) (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(2 a^2 C+b^2 (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{2 a b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}","-\frac{\sin (c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+b^2 (m+1) (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(2 a^2 C+b^2 (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{2 a b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}",1,"((2*a^2*C + b^2*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*(C*(2 + m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",6,5,33,0.1515,1,"{3050, 3033, 3023, 2748, 2643}"
767,1,217,0,0.3710767,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","-\frac{a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac{b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}","-\frac{a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac{b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}",1,"(a*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) + (b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)) - (a*(C*(1 + m) + A*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (b*(C*(2 + m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",5,4,31,0.1290,1,"{3034, 3023, 2748, 2643}"
768,1,353,0,0.456262,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{a \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}","\frac{a \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"(a*(A*b^2 + a^2*C)*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*AppellF1[1/2, -m/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/(b*(a^2 - b^2)*d*(Cos[c + d*x]^2)^(m/2)) + (a*C*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (C*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",8,5,33,0.1515,1,"{3064, 2643, 2823, 3189, 429}"
769,1,514,0,0.8807972,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sin (c+d x) \left(a^2 b^2 (A (-m)+A+C (m+2))+a^4 (-C) (m+1)+A b^4 m\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(a^2 b^2 (A (-m)+A+C (m+2))+a^4 (-C) (m+1)+A b^4 m\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(a^2 C (m+1)-b^2 (C-A m)\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{\sin (c+d x) \left(a^2 b^2 (A (-m)+A+C (m+2))+a^4 (-C) (m+1)+A b^4 m\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(a^2 b^2 (A (-m)+A+C (m+2))+a^4 (-C) (m+1)+A b^4 m\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(a^2 C (m+1)-b^2 (C-A m)\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((A*b^4*m - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)^2*d) - ((A*b^4*m - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, -m/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/(a*b*(a^2 - b^2)^2*d*(Cos[c + d*x]^2)^(m/2)) + ((A*b^2 + a^2*C)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((a^2*C*(1 + m) - b^2*(C - A*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) + ((A*b^2 + a^2*C)*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*b*(a^2 - b^2)*d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",9,6,33,0.1818,1,"{3056, 3063, 2643, 2823, 3189, 429}"
770,1,105,0,0.2076022,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{(a C+b B) \sin ^3(c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{(4 a B+3 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a B+3 b C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}","-\frac{(a C+b B) \sin ^3(c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{(4 a B+3 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a B+3 b C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"((4*a*B + 3*b*C)*x)/8 + ((b*B + a*C)*Sin[c + d*x])/d + ((4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((b*B + a*C)*Sin[c + d*x]^3)/(3*d)","A",8,7,36,0.1944,1,"{3029, 2968, 3023, 2748, 2635, 8, 2633}"
771,1,104,0,0.0806476,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(a^2 (-C)+3 a b B+2 b^2 C\right) \sin (c+d x)}{3 b d}+\frac{(3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} x (a C+b B)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}","\frac{(3 a B+2 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{b C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"((b*B + a*C)*x)/2 + ((3*a*b*B - a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)","A",2,2,30,0.06667,1,"{3023, 2734}"
772,1,52,0,0.065021,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (2 a B+b C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (2 a B+b C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*a*B + b*C)*x)/2 + ((b*B + a*C)*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",2,2,36,0.05556,1,"{3029, 2734}"
773,1,35,0,0.1590901,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","x (a C+b B)+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}","x (a C+b B)+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}",1,"(b*B + a*C)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d","A",5,5,38,0.1316,1,"{3029, 2968, 3023, 2735, 3770}"
774,1,35,0,0.171184,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+b C x","\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+b C x",1,"b*C*x + ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",5,5,38,0.1316,1,"{3029, 2968, 3021, 2735, 3770}"
775,1,61,0,0.1961379,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(a B+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(a B+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}",1,"((a*B + 2*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,38,0.1842,1,"{3029, 2968, 3021, 2748, 3767, 8, 3770}"
776,1,93,0,0.2367845,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{(2 a B+3 b C) \tan (c+d x)}{3 d}+\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{(2 a B+3 b C) \tan (c+d x)}{3 d}+\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"((b*B + a*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*B + 3*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",8,8,38,0.2105,1,"{3029, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
777,1,114,0,0.2270825,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{(a C+b B) \tan ^3(c+d x)}{3 d}+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a B+4 b C) \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 C+b B) \tan ^3(c+d x)}{3 d}+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a B+4 b C) \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,"((3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((b*B + a*C)*Tan[c + d*x])/d + ((3*a*B + 4*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((b*B + a*C)*Tan[c + d*x]^3)/(3*d)","A",8,7,38,0.1842,1,"{3029, 2968, 3021, 2748, 3767, 3768, 3770}"
778,1,189,0,0.3565174,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(4 a^2 B+6 a b C+3 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 B+6 a b C+3 b^2 B\right)-\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin (c+d x)}{5 d}+\frac{b (6 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}","\frac{\left(4 a^2 B+6 a b C+3 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 B+6 a b C+3 b^2 B\right)-\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin (c+d x)}{5 d}+\frac{b (6 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}",1,"((4*a^2*B + 3*b^2*B + 6*a*b*C)*x)/8 + ((4*b^2*C + 5*a*(2*b*B + a*C))*Sin[c + d*x])/(5*d) + ((4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*(5*b*B + 6*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (b*C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) - ((4*b^2*C + 5*a*(2*b*B + a*C))*Sin[c + d*x]^3)/(15*d)","A",8,7,38,0.1842,1,"{3029, 2990, 3023, 2748, 2635, 8, 2633}"
779,1,170,0,0.1930543,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 b d}+\frac{\left(-2 a^2 C+8 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 C+8 a b B+3 b^2 C\right)+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}","\frac{\left(4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 b d}+\frac{\left(-2 a^2 C+8 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 C+8 a b B+3 b^2 C\right)+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"((8*a*b*B + 4*a^2*C + 3*b^2*C)*x)/8 + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)","A",3,3,32,0.09375,1,"{3023, 2753, 2734}"
780,1,107,0,0.1592569,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \left(a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b C+b^2 B\right)+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{2 \left(a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b C+b^2 B\right)+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"((2*a^2*B + b^2*B + 2*a*b*C)*x)/2 + (2*(3*a*b*B + a^2*C + b^2*C)*Sin[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",3,3,38,0.07895,1,"{3029, 2753, 2734}"
781,1,86,0,0.2449155,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{1}{2} x \left(2 a^2 C+4 a b B+b^2 C\right)+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a C+2 b B) \sin (c+d x)}{2 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))}{2 d}","\frac{1}{2} x \left(2 a^2 C+4 a b B+b^2 C\right)+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a C+2 b B) \sin (c+d x)}{2 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"((4*a*b*B + 2*a^2*C + b^2*C)*x)/2 + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (b*(2*b*B + 3*a*C)*Sin[c + d*x])/(2*d) + (b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)","A",5,5,40,0.1250,1,"{3029, 2990, 3023, 2735, 3770}"
782,1,60,0,0.2427216,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 B \tan (c+d x)}{d}+\frac{a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac{b^2 C \sin (c+d x)}{d}","\frac{a^2 B \tan (c+d x)}{d}+\frac{a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac{b^2 C \sin (c+d x)}{d}",1,"b*(b*B + 2*a*C)*x + (a*(2*b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (b^2*C*Sin[c + d*x])/d + (a^2*B*Tan[c + d*x])/d","A",5,5,40,0.1250,1,"{3029, 2988, 3023, 2735, 3770}"
783,1,80,0,0.2795688,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(a^2 B+4 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a C+2 b B) \tan (c+d x)}{d}+b^2 C x","\frac{\left(a^2 B+4 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a C+2 b B) \tan (c+d x)}{d}+b^2 C x",1,"b^2*C*x + ((a^2*B + 2*b^2*B + 4*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*b*B + a*C)*Tan[c + d*x])/d + (a^2*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,40,0.1250,1,"{3029, 2988, 3021, 2735, 3770}"
784,1,116,0,0.3600391,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\left(2 a^2 B+6 a b C+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 C+2 a b B+2 b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec (c+d x)}{2 d}","\frac{\left(2 a^2 B+6 a b C+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 C+2 a b B+2 b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec (c+d x)}{2 d}",1,"((2*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a^2*B + 3*b^2*B + 6*a*b*C)*Tan[c + d*x])/(3*d) + (a*(2*b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,7,40,0.1750,1,"{3029, 2988, 3021, 2748, 3767, 8, 3770}"
785,1,156,0,0.3765418,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\left(2 a^2 C+4 a b B+3 b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{\left(2 a^2 C+4 a b B+3 b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"((3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*b*B + 2*a^2*C + 3*b^2*C)*Tan[c + d*x])/(3*d) + ((3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(2*b*B + a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a^2*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,8,40,0.2000,1,"{3029, 2988, 3021, 2748, 3768, 3770, 3767, 8}"
786,1,243,0,0.2934938,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 b d}+\frac{\left(-3 a^2 C+15 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right)+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}","\frac{\left(52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 b d}+\frac{\left(-3 a^2 C+15 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right)+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x)/8 + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)","A",4,3,32,0.09375,1,"{3023, 2753, 2734}"
787,1,171,0,0.2616764,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\left(16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{b \left(6 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right)+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{\left(16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{b \left(6 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right)+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"((8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*x)/8 + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sin[c + d*x])/(6*d) + (b*(20*a*b*B + 6*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",4,3,38,0.07895,1,"{3029, 2753, 2734}"
788,1,137,0,0.4791728,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b \left(8 a^2 C+9 a b B+2 b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(6 a^2 b B+2 a^3 C+3 a b^2 C+b^3 B\right)+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 (5 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{b \left(8 a^2 C+9 a b B+2 b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(6 a^2 b B+2 a^3 C+3 a b^2 C+b^3 B\right)+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 (5 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*x)/2 + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*b*B + 8*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*d) + (b^2*(3*b*B + 5*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{3029, 2990, 3033, 3023, 2735, 3770}"
789,1,131,0,0.4646385,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b \left(2 a^2 B-3 a b C-b^2 B\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+b^2 C\right)+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a B-b C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^2}{d}","-\frac{b \left(2 a^2 B-3 a b C-b^2 B\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+b^2 C\right)+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a B-b C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(b*(6*a*b*B + 6*a^2*C + b^2*C)*x)/2 + (a^2*(3*b*B + a*C)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a^2*B - b^2*B - 3*a*b*C)*Sin[c + d*x])/d - (b^2*(2*a*B - b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d","A",6,6,40,0.1500,1,"{3029, 2989, 3033, 3023, 2735, 3770}"
790,1,124,0,0.4132127,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \left(a^2 B+6 a b C+6 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}","\frac{a \left(a^2 B+6 a b C+6 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}",1,"b^2*(b*B + 3*a*C)*x + (a*(a^2*B + 6*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a*B - 2*b*C)*Sin[c + d*x])/(2*d) + (a^2*(2*b*B + a*C)*Tan[c + d*x])/d + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,40,0.1500,1,"{3029, 2989, 3031, 3023, 2735, 3770}"
791,1,145,0,0.429343,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \left(2 a^2 B+9 a b C+8 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 b B+a^3 C+6 a b^2 C+2 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 a C+5 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 C x","\frac{a \left(2 a^2 B+9 a b C+8 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 b B+a^3 C+6 a b^2 C+2 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 a C+5 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 C x",1,"b^3*C*x + ((3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2*B + 8*b^2*B + 9*a*b*C)*Tan[c + d*x])/(3*d) + (a^2*(5*b*B + 3*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{3029, 2989, 3031, 3021, 2735, 3770}"
792,1,188,0,0.5486357,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\left(6 a^2 b B+2 a^3 C+9 a b^2 C+3 b^3 B\right) \tan (c+d x)}{3 d}+\frac{\left(12 a^2 b C+3 a^3 B+12 a b^2 B+8 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(3 a^2 B+12 a b C+10 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a C+3 b B) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{\left(6 a^2 b B+2 a^3 C+9 a b^2 C+3 b^3 B\right) \tan (c+d x)}{3 d}+\frac{\left(12 a^2 b C+3 a^3 B+12 a b^2 B+8 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(3 a^2 B+12 a b C+10 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a C+3 b B) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"((3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*Tan[c + d*x])/(3*d) + (a*(3*a^2*B + 10*b^2*B + 12*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(3*b*B + 2*a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,8,40,0.2000,1,"{3029, 2989, 3031, 3021, 2748, 3767, 8, 3770}"
793,1,236,0,0.5616167,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{\left(30 a^2 b C+8 a^3 B+30 a b^2 B+15 b^3 C\right) \tan (c+d x)}{15 d}+\frac{\left(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(4 a^2 B+15 a b C+12 b^2 B\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (5 a C+7 b B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\left(30 a^2 b C+8 a^3 B+30 a b^2 B+15 b^3 C\right) \tan (c+d x)}{15 d}+\frac{\left(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(4 a^2 B+15 a b C+12 b^2 B\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (5 a C+7 b B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 15*b^3*C)*Tan[c + d*x])/(15*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(4*a^2*B + 12*b^2*B + 15*a*b*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(7*b*B + 5*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,9,40,0.2250,1,"{3029, 2989, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
794,1,178,0,0.5695806,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^3 (b B-a C) \tan ^{-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{x \left(2 a^2+b^2\right) (b B-a C)}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}","-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^3 (b B-a C) \tan ^{-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{x \left(2 a^2+b^2\right) (b B-a C)}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"((2*a^2 + b^2)*(b*B - a*C)*x)/(2*b^4) - (2*a^3*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",7,7,40,0.1750,1,"{3029, 2990, 3049, 3023, 2735, 2659, 205}"
795,1,134,0,0.3551281,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 a^2 (b B-a C) \tan ^{-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{x \left(-2 a^2 C+2 a b B-b^2 C\right)}{2 b^3}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}","\frac{2 a^2 (b B-a C) \tan ^{-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{x \left(-2 a^2 C+2 a b B-b^2 C\right)}{2 b^3}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"-((2*a*b*B - 2*a^2*C - b^2*C)*x)/(2*b^3) + (2*a^2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",6,6,38,0.1579,1,"{3029, 2990, 3023, 2735, 2659, 205}"
796,1,89,0,0.1312227,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","-\frac{2 a (b B-a C) \tan ^{-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{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}","-\frac{2 a (b B-a C) \tan ^{-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{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"((b*B - a*C)*x)/b^2 - (2*a*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)","A",5,5,32,0.1562,1,"{3023, 12, 2735, 2659, 205}"
797,1,67,0,0.1443575,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{C x}{b}","\frac{2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{C x}{b}",1,"(C*x)/b + (2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",4,4,38,0.1053,1,"{3029, 2735, 2659, 205}"
798,1,76,0,0.2080974,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (b B-a C) \tan ^{-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,"(-2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + (B*ArcTanh[Sin[c + d*x]])/(a*d)","A",5,5,40,0.1250,1,"{3029, 3001, 3770, 2659, 205}"
799,1,99,0,0.2729747,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{2 b (b B-a C) \tan ^{-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{(b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}","\frac{2 b (b B-a C) \tan ^{-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{(b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}",1,"(2*b*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - ((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) + (B*Tan[c + d*x])/(a*d)","A",7,7,40,0.1750,1,"{3029, 3000, 12, 2747, 3770, 2659, 205}"
800,1,143,0,0.5846067,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","-\frac{2 b^2 (b B-a C) \tan ^{-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{\left(a^2 B-2 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(b B-a C) \tan (c+d x)}{a^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 b^2 (b B-a C) \tan ^{-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{\left(a^2 B-2 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(b B-a C) \tan (c+d x)}{a^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(-2*b^2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) + ((a^2*B + 2*b^2*B - 2*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((b*B - a*C)*Tan[c + d*x])/(a^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",7,7,40,0.1750,1,"{3029, 3000, 3055, 3001, 3770, 2659, 205}"
801,1,263,0,0.7494194,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(2 a^2 b B-3 a^3 C+2 a b^2 C-b^3 B\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 b B-3 a^3 C+4 a b^2 C-3 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}-\frac{x \left(-6 a^2 C+4 a b B-b^2 C\right)}{2 b^4}","\frac{\left(2 a^2 b B-3 a^3 C+2 a b^2 C-b^3 B\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 b B-3 a^3 C+4 a b^2 C-3 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}-\frac{x \left(-6 a^2 C+4 a b B-b^2 C\right)}{2 b^4}",1,"-((4*a*b*B - 6*a^2*C - b^2*C)*x)/(2*b^4) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,40,0.1750,1,"{3029, 2989, 3049, 3023, 2735, 2659, 205}"
802,1,155,0,0.468126,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 a \left(a^2 b B-2 a^3 C+3 a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}","-\frac{2 a \left(a^2 b B-2 a^3 C+3 a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*x)/b^3 - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) - (a^2*(b*B - a*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,38,0.1579,1,"{3029, 2988, 3023, 2735, 2659, 205}"
803,1,122,0,0.1735996,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}","-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*x)/b^2 - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,32,0.1250,1,"{3021, 2735, 2659, 205}"
804,1,100,0,0.1536966,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{2 (a B-b C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{(b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 (a B-b C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{(b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*(a*B - b*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,38,0.1316,1,"{3029, 2754, 12, 2659, 205}"
805,1,133,0,0.385672,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(2 a^2 b B+a^3 (-C)-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}","-\frac{2 \left(2 a^2 b B+a^3 (-C)-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"(-2*(2*a^2*b*B - b^3*B - a^3*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (B*ArcTanh[Sin[c + d*x]])/(a^2*d) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,40,0.1500,1,"{3029, 3000, 3001, 3770, 2659, 205}"
806,1,189,0,0.7880057,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b \left(3 a^2 b B-2 a^3 C+a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 B+a b C-2 b^2 B\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{(2 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{2 b \left(3 a^2 b B-2 a^3 C+a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 B+a b C-2 b^2 B\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{(2 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(2*b*(3*a^2*b*B - 2*b^3*B - 2*a^3*C + a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((a^2*B - 2*b^2*B + a*b*C)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,40,0.1750,1,"{3029, 3000, 3055, 3001, 3770, 2659, 205}"
807,1,398,0,1.7654026,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-11 a^2 b^3 B+21 a^3 b^2 C+6 a^4 b B-12 a^5 C-6 a b^4 C+2 b^5 B\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-15 a^2 b^3 B+29 a^3 b^2 C+6 a^4 b B-12 a^5 C-20 a b^4 C+12 b^5 B\right) \tan ^{-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 (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(10 a^2 b^2 C+3 a^3 b B-6 a^4 C-6 a b^3 B-b^4 C\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}-\frac{x \left(-12 a^2 C+6 a b B-b^2 C\right)}{2 b^5}","\frac{\left(-11 a^2 b^3 B+21 a^3 b^2 C+6 a^4 b B-12 a^5 C-6 a b^4 C+2 b^5 B\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-15 a^2 b^3 B+29 a^3 b^2 C+6 a^4 b B-12 a^5 C-20 a b^4 C+12 b^5 B\right) \tan ^{-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 (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(10 a^2 b^2 C+3 a^3 b B-6 a^4 C-6 a b^3 B-b^4 C\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}-\frac{x \left(-12 a^2 C+6 a b B-b^2 C\right)}{2 b^5}",1,"-((6*a*b*B - 12*a^2*C - b^2*C)*x)/(2*b^5) + (a^2*(6*a^4*b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*C - 20*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,40,0.2000,1,"{3029, 2989, 3047, 3049, 3023, 2735, 2659, 205}"
808,1,280,0,1.2844089,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^2 b^3 B+15 a^3 b^2 C+2 a^4 b B-6 a^5 C-12 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a^2 \left(a^2 b B-3 a^3 C+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4}","-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^2 b^3 B+15 a^3 b^2 C+2 a^4 b B-6 a^5 C-12 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a^2 \left(a^2 b B-3 a^3 C+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4}",1,"((b*B - 3*a*C)*x)/b^4 - (a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C - 12*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*b*B - 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,40,0.1750,1,"{3029, 2989, 3031, 3023, 2735, 2659, 205}"
809,1,211,0,0.6152557,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-6 a b^4 C+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 (b B-a C) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2 b B-3 a^3 C+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{C x}{b^3}","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-6 a b^4 C+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 (b B-a C) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2 b B-3 a^3 C+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{C x}{b^3}",1,"(C*x)/b^3 + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 6*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(b*B - a*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,38,0.1579,1,"{3029, 2988, 3021, 2735, 2659, 205}"
810,1,180,0,0.2280817,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^2 b B+a^3 C-4 a b^2 C+2 b^3 B\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (b B-a C) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^2 b B+a^3 C-4 a b^2 C+2 b^3 B\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (b B-a C) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-(((3*a*b*B - a^2*C - 2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(b*B - a*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,32,0.1562,1,"{3021, 2754, 12, 2659, 205}"
811,1,164,0,0.2534872,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\left(2 a^2 B-3 a b C+b^2 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(b B-a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(2 a^2 B-3 a b C+b^2 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(b B-a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((2*a^2*B + b^2*B - 3*a*b*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((b*B - a*C)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*b*B - a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,5,38,0.1316,1,"{3029, 2754, 12, 2659, 205}"
812,1,214,0,0.8016406,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 C+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b \left(5 a^2 b B-3 a^3 C-2 b^3 B\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}","-\frac{\left(-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 C+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b \left(5 a^2 b B-3 a^3 C-2 b^3 B\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"-(((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + (B*ArcTanh[Sin[c + d*x]])/(a^3*d) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,40,0.1750,1,"{3029, 3000, 3055, 3001, 3770, 2659, 205}"
813,1,299,0,1.8034569,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{b \left(-15 a^2 b^3 B+5 a^3 b^2 C+12 a^4 b B-6 a^5 C-2 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(-11 a^2 b^2 B+5 a^3 b C+2 a^4 B-2 a b^3 C+6 b^4 B\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(6 a^2 b B-4 a^3 C+a b^2 C-3 b^3 B\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{(3 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}","\frac{b \left(-15 a^2 b^3 B+5 a^3 b^2 C+12 a^4 b B-6 a^5 C-2 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(-11 a^2 b^2 B+5 a^3 b C+2 a^4 B-2 a b^3 C+6 b^4 B\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(6 a^2 b B-4 a^3 C+a b^2 C-3 b^3 B\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{(3 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"(b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((3*b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((2*a^4*B - 11*a^2*b^2*B + 6*b^4*B + 5*a^3*b*C - 2*a*b^3*C)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(b*B - a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,7,40,0.1750,1,"{3029, 3000, 3055, 3001, 3770, 2659, 205}"
814,1,303,0,0.6048719,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 b B-8 a^3 C-19 a b^2 C-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}","-\frac{2 \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 b B-8 a^3 C-19 a b^2 C-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(-2*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - 19*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(14*a*b*B - 8*a^2*C - 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(14*a*b*B - 8*a^2*C - 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)","A",9,9,40,0.2250,1,"{3029, 2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
815,1,231,0,0.3463905,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 C+5 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 C+5 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,34,0.2059,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
816,1,171,0,0.3028794,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,7,40,0.1750,1,"{3029, 2753, 2752, 2663, 2661, 2655, 2653}"
817,1,178,0,0.4614935,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 b B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 b B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*b*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",9,9,42,0.2143,1,"{3029, 3002, 2655, 2653, 2803, 2663, 2661, 2807, 2805}"
818,1,213,0,0.7063219,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{(a B+2 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{(a B+2 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"-((B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a*B + 2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + ((b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",10,10,42,0.2381,1,"{3029, 2999, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
819,1,292,0,1.0474977,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(4 a^2 B+4 a b C-b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a C+b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","\frac{\left(4 a^2 B+4 a b C-b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a C+b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((3*b*B + 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B - b^2*B + 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,11,42,0.2619,1,"{3029, 2999, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
820,1,378,0,0.7887442,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \left(-8 a^2 C+18 a b B-49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-33 a^2 b^2 C+18 a^3 b B-8 a^4 C-246 a b^3 B-147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}","-\frac{2 \left(-8 a^2 C+18 a b B-49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(18 a^2 b B-8 a^3 C-39 a b^2 C-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-33 a^2 b^2 C+18 a^3 b B-8 a^4 C-246 a b^3 B-147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(-2*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 33*a^2*b^2*C - 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) - (2*(18*a*b*B - 8*a^2*C - 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)","A",10,9,40,0.2250,1,"{3029, 2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
821,1,297,0,0.4548376,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 b B-6 a^3 C+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}","\frac{2 \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 b B-6 a^3 C+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}",1,"(2*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,7,34,0.2059,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
822,1,225,0,0.4487747,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{2 \left(a^2-b^2\right) (3 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","-\frac{2 \left(a^2-b^2\right) (3 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B + 3*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",8,7,40,0.1750,1,"{3029, 2753, 2752, 2663, 2661, 2655, 2653}"
823,1,236,0,0.8330001,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 \left(a^2 (-C)+3 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a^2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (4 a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{2 \left(a^2 (-C)+3 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a^2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (4 a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*a*b*B - a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",10,10,42,0.2381,1,"{3029, 2990, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
824,1,232,0,0.8051161,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(a^2 B+2 a b C+2 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a B-2 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}","\frac{\left(a^2 B+2 a b C+2 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a B-2 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"-(((a*B - 2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a^2*B + 2*b^2*B + 2*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",10,10,42,0.2381,1,"{3029, 2989, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
825,1,295,0,1.1818533,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(4 a^2 C+7 a b B+8 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 B+12 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+5 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a C+5 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","\frac{\left(4 a^2 C+7 a b B+8 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 B+12 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+5 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a C+5 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((5*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((7*a*b*B + 4*a^2*C + 8*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B + 3*b^2*B + 12*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((5*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,11,42,0.2619,1,"{3029, 2989, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
826,1,375,0,1.5510457,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 B+42 a b C+17 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(12 a^2 b B+8 a^3 C+6 a b^2 C-b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 B+42 a b C+17 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(12 a^2 b B+8 a^3 C+6 a b^2 C-b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"-((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((16*a^2*B + 17*b^2*B + 42*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((7*b*B + 6*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",12,11,42,0.2619,1,"{3029, 2989, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
827,1,462,0,0.9798873,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \left(-8 a^2 C+22 a b B-81 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(110 a^2 b B-40 a^3 C-335 a b^2 C-539 b^3 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-3705 a b^4 C-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}","-\frac{2 \left(-8 a^2 C+22 a b B-81 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(110 a^2 b B-40 a^3 C-335 a b^2 C-539 b^3 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-3705 a b^4 C-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(-2*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 255*a^3*b^2*C - 3705*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 335*a*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*b*B - 8*a^2*C - 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)","A",11,9,40,0.2250,1,"{3029, 2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
828,1,372,0,0.6290956,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(-10 a^2 C+45 a b B+49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(45 a^2 b B-10 a^3 C+114 a b^2 C+75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(45 a^2 b B-10 a^3 C+114 a b^2 C+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(279 a^2 b^2 C+45 a^3 b B-10 a^4 C+435 a b^3 B+147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}","\frac{2 \left(-10 a^2 C+45 a b B+49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(45 a^2 b B-10 a^3 C+114 a b^2 C+75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(45 a^2 b B-10 a^3 C+114 a b^2 C+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(279 a^2 b^2 C+45 a^3 b B-10 a^4 C+435 a b^3 B+147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}",1,"(2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)","A",9,7,34,0.2059,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
829,1,288,0,0.5778265,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \left(15 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(15 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B+15 a^3 C+145 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}","\frac{2 \left(15 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(15 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B+15 a^3 C+145 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",9,7,40,0.1750,1,"{3029, 2753, 2752, 2663, 2661, 2655, 2653}"
830,1,292,0,1.1171493,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 \left(10 a^2 b B-8 a^3 C+8 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2 C+35 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 b (8 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","\frac{2 \left(10 a^2 b B-8 a^3 C+8 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2 C+35 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 b (8 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(35*a*b*B + 23*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C + 8*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*b*B + 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",11,11,42,0.2619,1,"{3029, 2990, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
831,1,296,0,1.1187046,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(4 a^2 b C+3 a^3 B+12 a b^2 B+2 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 B-14 a b C-6 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 a B-2 b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}","\frac{\left(4 a^2 b C+3 a^3 B+12 a b^2 B+2 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 B-14 a b C-6 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 a B-2 b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"-((3*a^2*B - 6*b^2*B - 14*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((3*a^3*B + 12*a*b^2*B + 4*a^2*b*C + 2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*(5*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*a*B - 2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",11,11,42,0.2619,1,"{3029, 2989, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
832,1,315,0,1.1473505,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(11 a^2 b B+4 a^3 C+16 a b^2 C+8 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 C+9 a b B-8 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 B+20 a b C+15 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a (4 a C+7 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}","\frac{\left(11 a^2 b B+4 a^3 C+16 a b^2 C+8 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 C+9 a b B-8 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 B+20 a b C+15 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a (4 a C+7 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"-((9*a*b*B + 4*a^2*C - 8*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((11*a^2*b*B + 8*b^3*B + 4*a^3*C + 16*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(4*a^2*B + 15*b^2*B + 20*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(7*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,11,42,0.2619,1,"{3029, 2989, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
833,1,376,0,1.5645512,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(66 a^2 b C+16 a^3 B+59 a b^2 B+48 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 b B+8 a^3 C+30 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a (2 a C+3 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(66 a^2 b C+16 a^3 B+59 a b^2 B+48 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 b B+8 a^3 C+30 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a (2 a C+3 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"-((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((16*a^3*B + 59*a*b^2*B + 66*a^2*b*C + 48*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(3*b*B + 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",12,12,42,0.2857,1,"{3029, 2989, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
834,1,465,0,1.9972547,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\left(284 a^2 b B+128 a^3 C+264 a b^2 C+15 b^3 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(356 a^2 b B+128 a^3 C+472 a b^2 C+133 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(284 a^2 b B+128 a^3 C+264 a b^2 C+15 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(120 a^2 b^2 B+160 a^3 b C+48 a^4 B+40 a b^3 C-5 b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(36 a^2 B+104 a b C+59 b^2 B\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{a (8 a C+11 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}","\frac{\left(284 a^2 b B+128 a^3 C+264 a b^2 C+15 b^3 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(356 a^2 b B+128 a^3 C+472 a b^2 C+133 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(284 a^2 b B+128 a^3 C+264 a b^2 C+15 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(120 a^2 b^2 B+160 a^3 b C+48 a^4 B+40 a b^3 C-5 b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(36 a^2 B+104 a b C+59 b^2 B\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{a (8 a C+11 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"-((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((356*a^2*b*B + 133*b^3*B + 128*a^3*C + 472*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((36*a^2*B + 59*b^2*B + 104*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*d) + (a*(11*b*B + 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",13,12,42,0.2857,1,"{3029, 2989, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
835,1,246,0,0.4865913,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(10 a^2 b B-8 a^3 C-7 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 C+10 a b B-9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}","\frac{2 \left(10 a^2 b B-8 a^3 C-7 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 C+10 a b B-9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(-2*(10*a*b*B - 8*a^2*C - 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - 7*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)","A",8,8,40,0.2000,1,"{3029, 2990, 3023, 2752, 2663, 2661, 2655, 2653}"
836,1,183,0,0.2222956,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \left(-2 a^2 C+3 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","-\frac{2 \left(-2 a^2 C+3 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a*b*B - 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,34,0.1765,1,"{3023, 2752, 2663, 2661, 2655, 2653}"
837,1,130,0,0.2371639,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,40,0.1500,1,"{3029, 2752, 2663, 2661, 2655, 2653}"
838,1,118,0,0.4013406,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,42,0.1429,1,"{3029, 3002, 2663, 2661, 2807, 2805}"
839,1,216,0,0.7198312,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{(b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{(b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"-((B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - ((b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)","A",10,10,42,0.2381,1,"{3029, 3000, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
840,1,299,0,1.1259164,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(4 a^2 B-4 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 b B-4 a C) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 b B-4 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(b B-4 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}","\frac{\left(4 a^2 B-4 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 b B-4 a C) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 b B-4 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(b B-4 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"((3*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((b*B - 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B + 3*b^2*B - 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",11,11,42,0.2619,1,"{3029, 3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
841,1,387,0,0.8748757,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-6 a^2 C+5 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(20 a^2 b B-24 a^3 C+9 a b^2 C-5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(40 a^2 b B-48 a^3 C-12 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(24 a^2 b^2 C+40 a^3 b B-48 a^4 C-25 a b^3 B+9 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-6 a^2 C+5 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(20 a^2 b B-24 a^3 C+9 a b^2 C-5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(40 a^2 b B-48 a^3 C-12 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(24 a^2 b^2 C+40 a^3 b B-48 a^4 C-25 a b^3 B+9 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 12*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*b*B - 6*a^2*C + b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",9,9,42,0.2143,1,"{3029, 2989, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
842,1,262,0,0.5672827,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 C+6 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 b B-8 a^3 C+5 a b^2 C-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 C+6 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 b B-8 a^3 C+5 a b^2 C-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(6*a*b*B - 8*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(b*B - a*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)","A",8,8,40,0.2000,1,"{3029, 2988, 3023, 2752, 2663, 2661, 2655, 2653}"
843,1,204,0,0.2822588,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,34,0.1765,1,"{3021, 2752, 2663, 2661, 2655, 2653}"
844,1,185,0,0.3360962,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"(2*(b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,40,0.1750,1,"{3029, 2754, 2752, 2663, 2661, 2655, 2653}"
845,1,190,0,0.64334,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,42,0.1905,1,"{3029, 3000, 3059, 2655, 2653, 12, 2807, 2805}"
846,1,303,0,1.1641827,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b \left(a^2 B+2 a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 B+2 a b C-3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}","\frac{b \left(a^2 B+2 a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 B+2 a b C-3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"-(((a^2*B - 3*b^2*B + 2*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(a^2*B - 3*b^2*B + 2*a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])","A",11,11,42,0.2619,1,"{3029, 3000, 3056, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
847,1,413,0,0.9506233,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 a^2 \left(3 a^2 b B-6 a^3 C+10 a b^2 C-7 b^3 B\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 \left(16 a^2 b^2 C+8 a^3 b B-16 a^4 C-9 a b^3 B+b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 b^3 B+28 a^3 b^2 C+8 a^4 b B-16 a^5 C-8 a b^4 C+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 a^2 \left(3 a^2 b B-6 a^3 C+10 a b^2 C-7 b^3 B\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 \left(16 a^2 b^2 C+8 a^3 b B-16 a^4 C-9 a b^3 B+b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 b^3 B+28 a^3 b^2 C+8 a^4 b B-16 a^5 C-8 a b^4 C+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 16*a^5*C + 28*a^3*b^2*C - 8*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*b*B - 9*a*b^3*B - 16*a^4*C + 16*a^2*b^2*C + b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a^2*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",9,9,42,0.2143,1,"{3029, 2989, 3031, 3023, 2752, 2663, 2661, 2655, 2653}"
848,1,331,0,0.6385879,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(2 a^2 b B-5 a^3 C+9 a b^2 C-6 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b B-8 a^3 C+9 a b^2 C-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(15 a^2 b^2 C+2 a^3 b B-8 a^4 C-6 a b^3 B-3 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(2 a^2 b B-5 a^3 C+9 a b^2 C-6 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 b B-8 a^3 C+9 a b^2 C-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(15 a^2 b^2 C+2 a^3 b B-8 a^4 C-6 a b^3 B-3 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*b*B - 3*b^3*B - 8*a^3*C + 9*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,40,0.2000,1,"{3029, 2988, 3021, 2752, 2663, 2661, 2655, 2653}"
849,1,307,0,0.4153133,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2 b B+2 a^3 C-6 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+a b B-3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b B+2 a^3 C-6 a b^2 C+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \left(a^2 b B+2 a^3 C-6 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+a b B-3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b B+2 a^3 C-6 a b^2 C+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a*b*B + 2*a^2*C - 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,34,0.2059,1,"{3021, 2754, 2752, 2663, 2661, 2655, 2653}"
850,1,275,0,0.4831735,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(4*a*b*B - a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(b*B - a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*b*B - a^2*C - 3*b^2*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,40,0.1750,1,"{3029, 2754, 2752, 2663, 2661, 2655, 2653}"
851,1,349,0,1.2497668,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b \left(7 a^2 b B-4 a^3 C-3 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(7 a^2 b B-4 a^3 C-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 b \left(7 a^2 b B-4 a^3 C-3 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(7 a^2 b B-4 a^3 C-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(b*B - a*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",11,11,42,0.2619,1,"{3029, 3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
852,1,437,0,1.6366634,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(-26 a^2 b^2 B+14 a^3 b C+3 a^4 B-6 a b^3 C+15 b^4 B\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 a^2 B+2 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(3 a^2 B+2 a b C-5 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-26 a^2 b^2 B+14 a^3 b C+3 a^4 B-6 a b^3 C+15 b^4 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}","\frac{b \left(-26 a^2 b^2 B+14 a^3 b C+3 a^4 B-6 a b^3 C+15 b^4 B\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 a^2 B+2 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(3 a^2 B+2 a b C-5 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-26 a^2 b^2 B+14 a^3 b C+3 a^4 B-6 a b^3 C+15 b^4 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"-((3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((3*a^2*B - 5*b^2*B + 2*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(3*a^2*B - 5*b^2*B + 2*a*b*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))","A",12,12,42,0.2857,1,"{3029, 3000, 3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
853,1,170,0,0.2671451,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a B+7 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a B+7 b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{10 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a B+7 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a B+7 b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*(9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",9,7,40,0.1750,1,"{3029, 2968, 3023, 2748, 2635, 2639, 2641}"
854,1,140,0,0.2399928,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 (7 a B+5 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a B+5 b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 (7 a B+5 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a B+5 b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(6*(b*B + a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",8,7,40,0.1750,1,"{3029, 2968, 3023, 2748, 2635, 2641, 2639}"
855,1,108,0,0.2193856,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a B+3 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a B+3 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",7,7,40,0.1750,1,"{3029, 2968, 3023, 2748, 2639, 2635, 2641}"
856,1,75,0,0.2049404,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 (3 a B+b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 (3 a B+b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(b*B + a*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a*B + b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{3029, 2968, 3023, 2748, 2641, 2639}"
857,1,71,0,0.210657,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a B-b C) 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 C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a B-b C) 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*B - b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/d + (2*a*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,40,0.1500,1,"{3029, 2968, 3021, 2748, 2641, 2639}"
858,1,103,0,0.2272164,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 (a B+3 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a C+b 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 B+3 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a C+b 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*(b*B + a*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,40,0.1750,1,"{3029, 2968, 3021, 2748, 2636, 2639, 2641}"
859,1,140,0,0.2476853,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a B+5 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a B+5 b C) \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 C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a B+5 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a B+5 b C) \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*(3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a*B + 5*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,40,0.1750,1,"{3029, 2968, 3021, 2748, 2636, 2641, 2639}"
860,1,264,0,0.4618286,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}","\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}",1,"(2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*C + 11*a*(2*b*B + a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 13*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)","A",9,7,42,0.1667,1,"{3029, 2990, 3023, 2748, 2635, 2639, 2641}"
861,1,223,0,0.4230992,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}","\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}",1,"(2*(7*b^2*C + 9*a*(2*b*B + a*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*b^2*C + 9*a*(2*b*B + a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 11*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)","A",8,7,42,0.1667,1,"{3029, 2990, 3023, 2748, 2635, 2641, 2639}"
862,1,182,0,0.3935845,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 a^2 B+6 a b C+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}","\frac{2 \left(5 a^2 B+6 a b C+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(2*(5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*b^2*C + 7*a*(2*b*B + a*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*b^2*C + 7*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 9*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",7,7,42,0.1667,1,"{3029, 2990, 3023, 2748, 2639, 2635, 2641}"
863,1,140,0,0.356083,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(3 a^2 B+2 a b C+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a C+2 b B)+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a C+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}","\frac{2 \left(3 a^2 B+2 a b C+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a C+2 b B)+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a C+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}",1,"(2*(3*b^2*C + 5*a*(2*b*B + a*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*(5*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",6,6,42,0.1429,1,"{3029, 2990, 3023, 2748, 2641, 2639}"
864,1,121,0,0.3354694,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(3 a^2 C+6 a b B+b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B-2 a b C-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 \left(3 a^2 C+6 a b B+b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B-2 a b C-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*(a^2*B - b^2*B - 2*a*b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*b*B + 3*a^2*C + b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,42,0.1429,1,"{3029, 2988, 3023, 2748, 2641, 2639}"
865,1,126,0,0.3597268,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(a^2 B+6 a b C+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 C+2 a b B-b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(a^2 B+6 a b C+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 C+2 a b B-b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*(2*a*b*B + a^2*C - b^2*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,42,0.1429,1,"{3029, 2988, 3021, 2748, 2641, 2639}"
866,1,172,0,0.3914975,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(a^2 C+2 a b B+3 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^2 C+2 a b B+3 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(2*a*b*B + a^2*C + 3*b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,7,42,0.1667,1,"{3029, 2988, 3021, 2748, 2636, 2639, 2641}"
867,1,214,0,0.4378185,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*B*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,42,0.1667,1,"{3029, 2988, 3021, 2748, 2636, 2641, 2639}"
868,1,305,0,0.638124,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(26 a^2 C+33 a b B+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}","\frac{2 \left(165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(26 a^2 C+33 a b B+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(27 a^2 b B+9 a^3 C+21 a b^2 C+7 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(165 a^2 b C+77 a^3 B+165 a b^2 B+45 b^3 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(2*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*b*B + 26*a^2*C + 9*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b^2*(11*b*B + 15*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)","A",9,8,42,0.1905,1,"{3029, 2990, 3033, 3023, 2748, 2635, 2641, 2639}"
869,1,255,0,0.5810753,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(22 a^2 C+27 a b B+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}","\frac{2 \left(21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(22 a^2 C+27 a b B+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(2*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*b*B + 22*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*b*B + 13*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)","A",8,8,42,0.1905,1,"{3029, 2990, 3033, 3023, 2748, 2639, 2635, 2641}"
870,1,205,0,0.5540933,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(21 a^2 b C+21 a^3 B+21 a b^2 B+5 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 b B+5 a^3 C+9 a b^2 C+3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(18 a^2 C+21 a b B+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}","\frac{2 \left(21 a^2 b C+21 a^3 B+21 a b^2 B+5 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 b B+5 a^3 C+9 a b^2 C+3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(18 a^2 C+21 a b B+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(2*(15*a^2*b*B + 3*b^3*B + 5*a^3*C + 9*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*b*B + 18*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*b*B + 11*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)","A",7,7,42,0.1667,1,"{3029, 2990, 3033, 3023, 2748, 2641, 2639}"
871,1,202,0,0.5584605,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(9 a^2 b B+3 a^3 C+3 a b^2 C+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(-15 a^2 b C+5 a^3 B-15 a b^2 B-3 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 B-3 a b C-b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}-\frac{2 b^2 (5 a B-b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(9 a^2 b B+3 a^3 C+3 a b^2 C+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(-15 a^2 b C+5 a^3 B-15 a b^2 B-3 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 B-3 a b C-b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}-\frac{2 b^2 (5 a B-b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}",1,"(-2*(5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a^2*B - b^2*B - 3*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*a*B - b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,42,0.1667,1,"{3029, 2989, 3033, 3023, 2748, 2641, 2639}"
872,1,192,0,0.5322731,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(9 a^2 b C+a^3 B+9 a b^2 B+b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (3 a C+7 b B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(9 a^2 b C+a^3 B+9 a b^2 B+b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (3 a C+7 b B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*b*B + 3*a*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(a*B - b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,7,42,0.1667,1,"{3029, 2989, 3031, 3023, 2748, 2641, 2639}"
873,1,204,0,0.5466129,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(3 a^2 b B+a^3 C+9 a b^2 C+3 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(15 a^2 b C+3 a^3 B+15 a b^2 B-5 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(3 a^2 B+15 a b C+14 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a C+9 b B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(3 a^2 b B+a^3 C+9 a b^2 C+3 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(15 a^2 b C+3 a^3 B+15 a b^2 B-5 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(3 a^2 B+15 a b C+14 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a C+9 b B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*b*B + 3*b^3*B + a^3*C + 9*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(9*b*B + 5*a*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*a^2*B + 14*b^2*B + 15*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,42,0.1667,1,"{3029, 2989, 3031, 3021, 2748, 2641, 2639}"
874,1,255,0,0.5950701,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(21 a^2 b C+5 a^3 B+21 a b^2 B+21 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2 B+21 a b C+18 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (7 a C+11 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(21 a^2 b C+5 a^3 B+21 a b^2 B+21 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2 B+21 a b C+18 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2 b B+3 a^3 C+15 a b^2 C+5 b^3 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (7 a C+11 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(11*b*B + 7*a*C)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*a^2*B + 18*b^2*B + 21*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,8,42,0.1905,1,"{3029, 2989, 3031, 3021, 2748, 2636, 2639, 2641}"
875,1,305,0,0.626176,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 \left(15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \left(7 a^2 B+27 a b C+22 b^2 B\right) \sin (c+d x)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (9 a C+13 b B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \left(15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \left(7 a^2 B+27 a b C+22 b^2 B\right) \sin (c+d x)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (9 a C+13 b B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(13*b*B + 9*a*C)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*a*(7*a^2*B + 22*b^2*B + 27*a*b*C)*Sin[c + d*x])/(45*d*Cos[c + d*x]^(5/2)) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",9,8,42,0.1905,1,"{3029, 2989, 3031, 3021, 2748, 2636, 2641, 2639}"
876,1,246,0,1.2039499,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(-7 a^2 b^2 C+21 a^3 b B-21 a^4 C+7 a b^3 B-5 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}+\frac{2 \left(5 a^2+3 b^2\right) (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}+\frac{2 a^4 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 \left(-7 a^2 C+7 a b B-5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}","-\frac{2 \left(-7 a^2 b^2 C+21 a^3 b B-21 a^4 C+7 a b^3 B-5 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}+\frac{2 \left(5 a^2+3 b^2\right) (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}+\frac{2 a^4 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 \left(-7 a^2 C+7 a b B-5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"(2*(5*a^2 + 3*b^2)*(b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*C - 5*b^4*C)*EllipticF[(c + d*x)/2, 2])/(21*b^5*d) + (2*a^4*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^5*(a + b)*d) - (2*(7*a*b*B - 7*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)","A",9,8,42,0.1905,1,"{3029, 2990, 3049, 3059, 2639, 3002, 2641, 2805}"
877,1,182,0,0.8989387,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(3 a^2+b^2\right) (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 C+5 a b B-3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a^3 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}","\frac{2 \left(3 a^2+b^2\right) (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 C+5 a b B-3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a^3 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"(-2*(5*a*b*B - 5*a^2*C - 3*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) + (2*(3*a^2 + b^2)*(b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(3*b^4*d) - (2*a^3*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^4*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)","A",8,8,42,0.1905,1,"{3029, 2990, 3049, 3059, 2639, 3002, 2641, 2805}"
878,1,137,0,0.6049836,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(-3 a^2 C+3 a b B-b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","-\frac{2 \left(-3 a^2 C+3 a b B-b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(2*(b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(b^2*d) - (2*(3*a*b*B - 3*a^2*C - b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) + (2*a^2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",7,7,42,0.1667,1,"{3029, 2990, 3059, 2639, 3002, 2641, 2805}"
879,1,89,0,0.2984498,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*(b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(b^2*d) - (2*a*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",6,6,42,0.1429,1,"{3029, 3002, 2639, 2803, 2641, 2805}"
880,1,61,0,0.2421937,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*EllipticF[(c + d*x)/2, 2])/(b*d) + (2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)","A",4,4,42,0.09524,1,"{3029, 3002, 2641, 2805}"
881,1,86,0,0.4088867,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 B \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","-\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 B \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(-2*B*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d) + (2*B*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])","A",6,6,42,0.1429,1,"{3029, 3000, 3059, 2639, 12, 2805}"
882,1,150,0,0.8719814,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (b B-a C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (b B-a C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*b*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d) + (2*B*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(b*B - a*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])","A",8,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
883,1,217,0,1.2360636,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 (b B-a C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 (b B-a C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*(b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (2*b^2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*B*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(b*B - a*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])","A",9,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
884,1,389,0,1.3822246,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(-16 a^2 b^3 B+20 a^3 b^2 C+15 a^4 b B-21 a^5 C+4 a b^4 C-2 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{\left(24 a^2 b^2 C+25 a^3 b B-35 a^4 C-20 a b^3 B+6 b^4 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^3 \left(5 a^2 b B-7 a^3 C+9 a b^2 C-7 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-7 a^2 C+5 a b B+2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\left(5 a^2 b B-7 a^3 C+4 a b^2 C-2 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}","\frac{\left(-16 a^2 b^3 B+20 a^3 b^2 C+15 a^4 b B-21 a^5 C+4 a b^4 C-2 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{\left(24 a^2 b^2 C+25 a^3 b B-35 a^4 C-20 a b^3 B+6 b^4 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^3 \left(5 a^2 b B-7 a^3 C+9 a b^2 C-7 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-7 a^2 C+5 a b B+2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\left(5 a^2 b B-7 a^3 C+4 a b^2 C-2 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}",1,"-((25*a^3*b*B - 20*a*b^3*B - 35*a^4*C + 24*a^2*b^2*C + 6*b^4*C)*EllipticE[(c + d*x)/2, 2])/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - 21*a^5*C + 20*a^3*b^2*C + 4*a*b^4*C)*EllipticF[(c + d*x)/2, 2])/(3*b^5*(a^2 - b^2)*d) - (a^3*(5*a^2*b*B - 7*b^3*B - 7*a^3*C + 9*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^5*(a + b)^2*d) + ((5*a^2*b*B - 2*b^3*B - 7*a^3*C + 4*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) - ((5*a*b*B - 7*a^2*C + 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",9,8,42,0.1905,1,"{3029, 2989, 3049, 3059, 2639, 3002, 2641, 2805}"
885,1,303,0,1.0245151,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\left(16 a^2 b^2 C+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b B-5 a^3 C+4 a b^2 C-2 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 b B-5 a^3 C+7 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-5 a^2 C+3 a b B+2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}","-\frac{\left(16 a^2 b^2 C+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b B-5 a^3 C+4 a b^2 C-2 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 b B-5 a^3 C+7 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-5 a^2 C+3 a b B+2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}",1,"((3*a^2*b*B - 2*b^3*B - 5*a^3*C + 4*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - 15*a^4*C + 16*a^2*b^2*C + 2*b^4*C)*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*b*B - 5*b^3*B - 5*a^3*C + 7*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) - ((3*a*b*B - 5*a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 2989, 3049, 3059, 2639, 3002, 2641, 2805}"
886,1,224,0,0.7047207,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 b B-3 a^3 C+4 a b^2 C-2 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 b B-3 a^3 C+5 a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{\left(a^2 b B-3 a^3 C+4 a b^2 C-2 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 b B-3 a^3 C+5 a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}+\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-(((a*b*B - 3*a^2*C + 2*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d)) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - (a*(a^2*b*B - 3*b^3*B - 3*a^3*C + 5*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) + (a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,42,0.1667,1,"{3029, 2989, 3059, 2639, 3002, 2641, 2805}"
887,1,198,0,0.6359304,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\left(a^2 C+a b B-2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 b B+a^3 C-3 a b^2 C+b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{\left(a^2 C+a b B-2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 b B+a^3 C-3 a b^2 C+b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((a*b*B + a^2*C - 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - ((a^2*b*B + b^3*B + a^3*C - 3*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) - ((b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,42,0.1667,1,"{3029, 2999, 3059, 2639, 3002, 2641, 2805}"
888,1,200,0,0.7138926,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","-\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) 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 b B+a^3 (-C)-a b^2 C-b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}+\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) 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 b B+a^3 (-C)-a b^2 C-b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}+\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-(((b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d)) - ((b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - a^3*C - a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b*(a + b)^2*d) + (b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,42,0.1667,1,"{3029, 3000, 3059, 2639, 3002, 2641, 2805}"
889,1,256,0,1.0126333,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 B+a b C-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 b B-3 a^3 C+a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}+\frac{\left(2 a^2 B+a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}","\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 B+a b C-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 b B-3 a^3 C+a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}+\frac{\left(2 a^2 B+a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}",1,"-(((2*a^2*B - 3*b^2*B + a*b*C)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d)) + ((b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) - ((5*a^2*b*B - 3*b^3*B - 3*a^3*C + a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*B - 3*b^2*B + a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
890,1,345,0,1.3846016,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(4 a^2 b B-2 a^3 C+3 a b^2 C-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 b B-5 a^3 C+3 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(4 a^2 b B-2 a^3 C+3 a b^2 C-5 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(4 a^2 b B-2 a^3 C+3 a b^2 C-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 b B-5 a^3 C+3 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(4 a^2 b B-2 a^3 C+3 a b^2 C-5 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*b*B - 5*b^3*B - 5*a^3*C + 3*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",9,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
891,1,461,0,1.5446983,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-99 a^3 b^3 B+223 a^4 b^2 C-128 a^2 b^4 C+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 b^3 B+65 a^3 b^2 C+15 a^4 b B-35 a^5 C-24 a b^4 C+8 b^5 B\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^2 \left(-38 a^2 b^3 B+86 a^3 b^2 C+15 a^4 b B-35 a^5 C-63 a b^4 C+35 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(3 a^2 b B-7 a^3 C+13 a b^2 C-9 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(61 a^2 b^2 C+15 a^3 b B-35 a^4 C-33 a b^3 B-8 b^4 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}","-\frac{\left(-99 a^3 b^3 B+223 a^4 b^2 C-128 a^2 b^4 C+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 b^3 B+65 a^3 b^2 C+15 a^4 b B-35 a^5 C-24 a b^4 C+8 b^5 B\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^2 \left(-38 a^2 b^3 B+86 a^3 b^2 C+15 a^4 b B-35 a^5 C-63 a b^4 C+35 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(3 a^2 b B-7 a^3 C+13 a b^2 C-9 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(61 a^2 b^2 C+15 a^3 b B-35 a^4 C-33 a b^3 B-8 b^4 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}",1,"((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - 35*a^5*C + 65*a^3*b^2*C - 24*a*b^4*C)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - 105*a^6*C + 223*a^4*b^2*C - 128*a^2*b^4*C - 8*b^6*C)*EllipticF[(c + d*x)/2, 2])/(12*b^5*(a^2 - b^2)^2*d) + (a^2*(15*a^4*b*B - 38*a^2*b^3*B + 35*b^5*B - 35*a^5*C + 86*a^3*b^2*C - 63*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*b*B - 33*a*b^3*B - 35*a^4*C + 61*a^2*b^2*C - 8*b^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(3*a^2*b*B - 9*b^3*B - 7*a^3*C + 13*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",9,9,42,0.2143,1,"{3029, 2989, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
892,1,367,0,1.1000877,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-5 a^2 b^3 B+33 a^3 b^2 C+3 a^4 b B-15 a^5 C-24 a b^4 C+8 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(29 a^2 b^2 C+3 a^3 b B-15 a^4 C-9 a b^3 B-8 b^4 C\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{a \left(-6 a^2 b^3 B+38 a^3 b^2 C+3 a^4 b B-15 a^5 C-35 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2 b B-5 a^3 C+11 a b^2 C-7 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}","\frac{\left(-5 a^2 b^3 B+33 a^3 b^2 C+3 a^4 b B-15 a^5 C-24 a b^4 C+8 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(29 a^2 b^2 C+3 a^3 b B-15 a^4 C-9 a b^3 B-8 b^4 C\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{a \left(-6 a^2 b^3 B+38 a^3 b^2 C+3 a^4 b B-15 a^5 C-35 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}+\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2 b B-5 a^3 C+11 a b^2 C-7 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"-((3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*b*B - 6*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 38*a^3*b^2*C - 35*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b*B - 7*b^3*B - 5*a^3*C + 11*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 2989, 3047, 3059, 2639, 3002, 2641, 2805}"
893,1,344,0,1.097232,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-5 a^2 b^2 C+a^3 b B+3 a^4 C-7 a b^3 B+8 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b B+3 a^3 C-9 a b^2 C+5 b^3 B\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 b^3 B-6 a^3 b^2 C+a^4 b B+3 a^5 C+15 a b^4 C-3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}+\frac{\left(a^2 b B+3 a^3 C-9 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(-5 a^2 b^2 C+a^3 b B+3 a^4 C-7 a b^3 B+8 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b B+3 a^3 C-9 a b^2 C+5 b^3 B\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 b^3 B-6 a^3 b^2 C+a^4 b B+3 a^5 C+15 a b^4 C-3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}+\frac{\left(a^2 b B+3 a^3 C-9 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*b*B - 10*a^2*b^3*B - 3*b^5*B + 3*a^5*C - 6*a^3*b^2*C + 15*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 2989, 3055, 3059, 2639, 3002, 2641, 2805}"
894,1,337,0,1.0256853,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\left(3 a^2 b B+a^3 C-7 a b^2 C+3 b^3 B\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(5 a^2 b B+a^3 (-C)-5 a b^2 C+b^3 B\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 b^3 B-10 a^3 b^2 C+3 a^4 b B+a^5 C-3 a b^4 C-b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d (a-b)^2 (a+b)^3}-\frac{\left(5 a^2 b B+a^3 (-C)-5 a b^2 C+b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(3 a^2 b B+a^3 C-7 a b^2 C+3 b^3 B\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(5 a^2 b B+a^3 (-C)-5 a b^2 C+b^3 B\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 b^3 B-10 a^3 b^2 C+3 a^4 b B+a^5 C-3 a b^4 C-b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d (a-b)^2 (a+b)^3}-\frac{\left(5 a^2 b B+a^3 (-C)-5 a b^2 C+b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*b*B + 3*b^3*B + a^3*C - 7*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*b*B + 10*a^2*b^3*B - b^5*B + a^5*C - 10*a^3*b^2*C - 3*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 2999, 3055, 3059, 2639, 3002, 2641, 2805}"
895,1,345,0,1.1718653,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","-\frac{\left(7 a^2 b B-3 a^3 C-3 a b^2 C-b^3 B\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(9 a^2 b B-5 a^3 C-a b^2 C-3 b^3 B\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 b^3 B-10 a^3 b^2 C+15 a^4 b B-3 a^5 C+a b^4 C+3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d (a-b)^2 (a+b)^3}+\frac{b \left(9 a^2 b B-5 a^3 C-a b^2 C-3 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(7 a^2 b B-3 a^3 C-3 a b^2 C-b^3 B\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(9 a^2 b B-5 a^3 C-a b^2 C-3 b^3 B\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 b^3 B-10 a^3 b^2 C+15 a^4 b B-3 a^5 C+a b^4 C+3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d (a-b)^2 (a+b)^3}+\frac{b \left(9 a^2 b B-5 a^3 C-a b^2 C-3 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*b*B - b^3*B - 3*a^3*C - 3*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*b*B - 6*a^2*b^3*B + 3*b^5*B - 3*a^5*C - 10*a^3*b^2*C + a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
896,1,420,0,1.5754842,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\left(11 a^2 b B-7 a^3 C+a b^2 C-5 b^3 B\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(-29 a^2 b^2 B+9 a^3 b C+8 a^4 B-3 a b^3 C+15 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 b^3 B+6 a^3 b^2 C+35 a^4 b B-15 a^5 C-3 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}+\frac{\left(-29 a^2 b^2 B+9 a^3 b C+8 a^4 B-3 a b^3 C+15 b^4 B\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{b \left(11 a^2 b B-7 a^3 C+a b^2 C-5 b^3 B\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}","\frac{\left(11 a^2 b B-7 a^3 C+a b^2 C-5 b^3 B\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(-29 a^2 b^2 B+9 a^3 b C+8 a^4 B-3 a b^3 C+15 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 b^3 B+6 a^3 b^2 C+35 a^4 b B-15 a^5 C-3 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}+\frac{\left(-29 a^2 b^2 B+9 a^3 b C+8 a^4 B-3 a b^3 C+15 b^4 B\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{b \left(11 a^2 b B-7 a^3 C+a b^2 C-5 b^3 B\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}",1,"-((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*b*B - 38*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 6*a^3*b^2*C - 3*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b*(11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",9,8,42,0.1905,1,"{3029, 3000, 3055, 3059, 2639, 3002, 2641, 2805}"
897,1,560,0,1.6590773,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(-3 a^2 C+6 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+6 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(2 a^2 b B+a^3 (-C)-4 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sqrt{a+b} (a+2 b) (-3 a C+6 b B+8 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}","\frac{\left(-3 a^2 C+6 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+6 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(2 a^2 b B+a^3 (-C)-4 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sqrt{a+b} (a+2 b) (-3 a C+6 b B+8 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}",1,"-((a - b)*Sqrt[a + b]*(6*a*b*B - 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) + (Sqrt[a + b]*(a + 2*b)*(6*b*B - 3*a*C + 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) + ((6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)","A",9,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
898,1,473,0,1.1741312,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \left(a^2 (-C)+4 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a C+2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\sqrt{a+b} \left(a^2 (-C)+4 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a C+2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(a*C + 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(4*a*b*B - a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",8,8,44,0.1818,1,"{3029, 3003, 3061, 3053, 2809, 2998, 2816, 2994}"
899,1,385,0,0.8423221,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{a+b} (2 B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}","\frac{\sqrt{a+b} (2 B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"-(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)) + (Sqrt[a + b]*(2*B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*b*B + a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,44,0.1591,1,"{3029, 3003, 3053, 2809, 2998, 2816, 2994}"
900,1,351,0,0.6407276,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b} (b B-a (B-C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \sqrt{a+b} (b B-a (B-C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (2*Sqrt[a + b]*(b*B - a*(B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d","A",6,6,44,0.1364,1,"{3029, 2991, 2809, 2998, 2816, 2994}"
901,1,284,0,0.6271707,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} (3 a C+b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 (a-b) \sqrt{a+b} (B-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (a-b) \sqrt{a+b} (3 a C+b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 (a-b) \sqrt{a+b} (B-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(b*B + 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*(a - b)*Sqrt[a + b]*(B - 3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",5,5,44,0.1136,1,"{3029, 2999, 2998, 2816, 2994}"
902,1,350,0,0.9521104,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+5 a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+5 a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2*B - 2*b^2*B + 5*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a*B + 2*b*B - 5*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))","A",6,6,44,0.1364,1,"{3029, 2999, 3055, 2998, 2816, 2994}"
903,1,433,0,1.291212,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(25 a^2 B+7 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)+2 a b (3 B-7 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(19 a^2 b B+63 a^3 C-14 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(25 a^2 B+7 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)+2 a b (3 B-7 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(19 a^2 b B+63 a^3 C-14 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(19*a^2*b*B + 8*b^3*B + 63*a^3*C - 14*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*b^2*B + a^2*(25*B - 63*C) + 2*a*b*(3*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B - 4*b^2*B + 7*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))","A",7,6,44,0.1364,1,"{3029, 2999, 3055, 2998, 2816, 2994}"
904,1,670,0,2.2730269,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(-3 a^2 C+8 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{\left(24 a^2 b B-9 a^3 C+156 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (28 B+39 C)-8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 b B-9 a^3 C+156 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \left(-24 a^2 b^2 C+8 a^3 b B-3 a^4 C-96 a b^3 B-48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}","\frac{\left(-3 a^2 C+8 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{\left(24 a^2 b B-9 a^3 C+156 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (28 B+39 C)-8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 b B-9 a^3 C+156 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \left(-24 a^2 b^2 C+8 a^3 b B-3 a^4 C-96 a b^3 B-48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}",1,"-((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(16*B + 9*C) - 4*a*b^2*(28*B + 39*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*C - 48*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((8*a*b*B - 3*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)","A",10,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
905,1,566,0,1.8576249,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\left(3 a^2 C+30 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+30 a b B+14 a b C+12 b^2 B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+30 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \left(6 a^2 b B+a^3 (-C)+12 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(7 a C+6 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{\left(3 a^2 C+30 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+30 a b B+14 a b C+12 b^2 B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+30 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \left(6 a^2 b B+a^3 (-C)+12 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(7 a C+6 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"-((a - b)*Sqrt[a + b]*(30*a*b*B + 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d) + (Sqrt[a + b]*(30*a*b*B + 12*b^2*B + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((6*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
906,1,472,0,1.3167215,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{\sqrt{a+b} \left(3 a^2 C+12 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(5 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (8 a B+5 a C+4 b B+2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\sqrt{a+b} \left(3 a^2 C+12 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(5 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (8 a B+5 a C+4 b B+2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((a - b)*Sqrt[a + b]*(4*b*B + 5*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) + (Sqrt[a + b]*(8*a*B + 4*b*B + 5*a*C + 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(12*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + ((4*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",8,8,44,0.1818,1,"{3029, 2990, 3061, 3053, 2809, 2998, 2816, 2994}"
907,1,449,0,1.3049857,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{(2 a B-b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (B-C)-b (4 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} (2 a B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","-\frac{(2 a B-b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (B-C)-b (4 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} (2 a B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(2*a*B - b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a*(B - C) - b*(4*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*b*B + 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a*B - b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,8,44,0.1818,1,"{3029, 2989, 3061, 3053, 2809, 2998, 2816, 2994}"
908,1,418,0,0.9943441,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \sqrt{a+b} \left(a^2 (B-3 C)-a b (4 B-6 C)+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a-b) \sqrt{a+b} (3 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \sqrt{a+b} \left(a^2 (B-3 C)-a b (4 B-6 C)+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a-b) \sqrt{a+b} (3 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*(4*b*B + 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*Sqrt[a + b]*(3*b^2*B - a*b*(4*B - 6*C) + a^2*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,7,44,0.1591,1,"{3029, 2989, 3053, 2809, 2998, 2816, 2994}"
909,1,353,0,1.0507036,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+20 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a C+6 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C-3 b B+15 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+20 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a C+6 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C-3 b B+15 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2*B + 3*b^2*B + 20*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) - (2*(a - b)*Sqrt[a + b]*(9*a*B - 3*b*B - 5*a*C + 15*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",6,6,44,0.1364,1,"{3029, 2989, 3055, 2998, 2816, 2994}"
910,1,433,0,1.4174006,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(25 a^2 B+42 a b C+3 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(a^2 (-(25 B-63 C))+3 a b (19 B-7 C)+6 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(82 a^2 b B+63 a^3 C+21 a b^2 C-6 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(25 a^2 B+42 a b C+3 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(a^2 (-(25 B-63 C))+3 a b (19 B-7 C)+6 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(82 a^2 b B+63 a^3 C+21 a b^2 C-6 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(82*a^2*b*B - 6*b^3*B + 63*a^3*C + 21*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*b^2*B - a^2*(25*B - 63*C) + 3*a*b*(19*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(8*b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B + 3*b^2*B + 42*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))","A",7,6,44,0.1364,1,"{3029, 2989, 3055, 2998, 2816, 2994}"
911,1,779,0,3.2902647,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\left(-15 a^2 C+50 a b B+64 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(50 a^2 b B-15 a^3 C+172 a b^2 C+120 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-4 a^2 b^2 (295 B+423 C)-30 a^3 b (5 B+C)+45 a^4 C-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-240 a^2 b^3 B-40 a^3 b^2 C+10 a^4 b B-3 a^5 C-240 a b^4 C-96 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}+\frac{(10 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}","\frac{\left(-15 a^2 C+50 a b B+64 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(50 a^2 b B-15 a^3 C+172 a b^2 C+120 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-4 a^2 b^2 (295 B+423 C)-30 a^3 b (5 B+C)+45 a^4 C-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(1692 a^2 b^2 C+150 a^3 b B-45 a^4 C+2840 a b^3 B+1024 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-240 a^2 b^3 B-40 a^3 b^2 C+10 a^4 b B-3 a^5 C-240 a b^4 C-96 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}+\frac{(10 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}",1,"-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(45*B + 64*C) - 8*a*b^3*(355*B + 193*C) - 4*a^2*b^2*(295*B + 423*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*C - 240*a*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 172*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)","A",11,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
912,1,664,0,2.3610717,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\left(5 a^2 C+24 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(264 a^2 b B+15 a^3 C+284 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (132 B+59 C)+15 a^3 C+4 a b^2 (52 B+71 C)+8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(264 a^2 b B+15 a^3 C+284 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \left(120 a^2 b^2 C+40 a^3 b B-5 a^4 C+160 a b^3 B+48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(11 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}","\frac{\left(5 a^2 C+24 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(264 a^2 b B+15 a^3 C+284 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (132 B+59 C)+15 a^3 C+4 a b^2 (52 B+71 C)+8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(264 a^2 b B+15 a^3 C+284 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \left(120 a^2 b^2 C+40 a^3 b B-5 a^4 C+160 a b^3 B+48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(11 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"-((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(16*B + 9*C) + 2*a^2*b*(132*B + 59*C) + 4*a*b^2*(52*B + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",10,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
913,1,563,0,1.8242783,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\left(33 a^2 C+54 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (48 B+33 C)+a b (54 B+26 C)+4 b^2 (3 B+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 C+54 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \left(30 a^2 b B+5 a^3 C+20 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b (3 a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\left(33 a^2 C+54 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (48 B+33 C)+a b (54 B+26 C)+4 b^2 (3 B+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 C+54 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \left(30 a^2 b B+5 a^3 C+20 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b (3 a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"-((a - b)*Sqrt[a + b]*(54*a*b*B + 33*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) + (Sqrt[a + b]*(4*b^2*(3*B + 4*C) + a*b*(54*B + 26*C) + a^2*(48*B + 33*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((54*a*b*B + 33*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*(2*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
914,1,547,0,1.7850334,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{\left(8 a^2 B-9 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(8 a^2 (B-C)-3 a b (8 B+3 C)-2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 B-9 a b C-4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 C+20 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b (4 a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}","-\frac{\left(8 a^2 B-9 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(8 a^2 (B-C)-3 a b (8 B+3 C)-2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 B-9 a b C-4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 C+20 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b (4 a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(8*a^2*B - 4*b^2*B - 9*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) - (Sqrt[a + b]*(8*a^2*(B - C) - 2*b^2*(2*B + C) - 3*a*b*(8*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(20*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - ((8*a^2*B - 4*b^2*B - 9*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",9,9,44,0.2045,1,"{3029, 2989, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
915,1,536,0,1.7602193,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{\left(6 a^2 C+14 a b B-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 (B-3 C)+2 a b (7 B-9 C)-3 b^2 (6 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 C+14 a b B-3 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a (a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{\left(6 a^2 C+14 a b B-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 (B-3 C)+2 a b (7 B-9 C)-3 b^2 (6 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 C+14 a b B-3 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a (a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((a - b)*Sqrt[a + b]*(14*a*b*B + 6*a^2*C - 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (Sqrt[a + b]*(2*a*b*(7*B - 9*C) - 2*a^2*(B - 3*C) - 3*b^2*(6*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(2*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((14*a*b*B + 6*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",9,9,44,0.2045,1,"{3029, 2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
916,1,493,0,1.3618354,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \sqrt{a+b} \left(a^2 b (17 B-35 C)+a^3 (-(9 B-5 C))-a b^2 (23 B-45 C)+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+35 a b C+23 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a (5 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \sqrt{a+b} \left(a^2 b (17 B-35 C)+a^3 (-(9 B-5 C))-a b^2 (23 B-45 C)+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+35 a b C+23 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a (5 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2*B + 23*b^2*B + 35*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*Sqrt[a + b]*(15*b^3*B - a*b^2*(23*B - 45*C) + a^2*b*(17*B - 35*C) - a^3*(9*B - 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(8*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,8,44,0.1818,1,"{3029, 2989, 3047, 3053, 2809, 2998, 2816, 2994}"
917,1,434,0,1.4523279,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(25 a^2 B+77 a b C+45 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)-8 a b (15 B-7 C)+15 b^2 (B-7 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 b B+63 a^3 C+161 a b^2 C+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 a (7 a C+10 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(25 a^2 B+77 a b C+45 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)-8 a b (15 B-7 C)+15 b^2 (B-7 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 b B+63 a^3 C+161 a b^2 C+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 a (7 a C+10 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(145*a^2*b*B + 15*b^3*B + 63*a^3*C + 161*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*B - 63*C) + 15*b^2*(B - 7*C) - 8*a*b*(15*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d) + (2*a*(10*b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B + 45*b^2*B + 77*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,7,44,0.1591,1,"{3029, 2989, 3047, 3055, 2998, 2816, 2994}"
918,1,522,0,1.955317,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 \left(163 a^2 b B+75 a^3 C+135 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2 B+135 a b C+75 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 B-60 C)+3 a^3 (49 B-25 C)+15 a b^2 (11 B-3 C)+10 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(279 a^2 b^2 B+435 a^3 b C+147 a^4 B+45 a b^3 C-10 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 a (3 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \left(163 a^2 b B+75 a^3 C+135 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2 B+135 a b C+75 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 B-60 C)+3 a^3 (49 B-25 C)+15 a b^2 (11 B-3 C)+10 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(279 a^2 b^2 B+435 a^3 b C+147 a^4 B+45 a b^3 C-10 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 a (3 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(147*a^4*B + 279*a^2*b^2*B - 10*b^4*B + 435*a^3*b*C + 45*a*b^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*b^3*B - 6*a^2*b*(19*B - 60*C) + 3*a^3*(49*B - 25*C) + 15*a*b^2*(11*B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*a*(4*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2*B + 75*b^2*B + 135*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(163*a^2*b*B + 5*b^3*B + 75*a^3*C + 135*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,7,44,0.1591,1,"{3029, 2989, 3047, 3055, 2998, 2816, 2994}"
919,1,622,0,2.7531518,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{2 \left(1025 a^2 b^2 B+1793 a^3 b C+675 a^4 B+55 a b^3 C-20 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(1145 a^2 b B+539 a^3 C+825 a b^2 C+15 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(81 a^2 B+209 a b C+113 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^2 b^2 (19 B-121 C)-6 a^3 b (505 B-209 C)+3 a^4 (225 B-539 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 b^3 B+3069 a^3 b^2 C+3705 a^4 b B+1617 a^5 C-110 a b^4 C+40 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d}+\frac{2 a (11 a C+14 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 \left(1025 a^2 b^2 B+1793 a^3 b C+675 a^4 B+55 a b^3 C-20 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(1145 a^2 b B+539 a^3 C+825 a b^2 C+15 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(81 a^2 B+209 a b C+113 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^2 b^2 (19 B-121 C)-6 a^3 b (505 B-209 C)+3 a^4 (225 B-539 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 b^3 B+3069 a^3 b^2 C+3705 a^4 b B+1617 a^5 C-110 a b^4 C+40 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d}+\frac{2 a (11 a C+14 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(3705*a^4*b*B + 255*a^2*b^3*B + 40*b^5*B + 1617*a^5*C + 3069*a^3*b^2*C - 110*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d) + (2*(a - b)*Sqrt[a + b]*(40*b^4*B + 3*a^4*(225*B - 539*C) - 6*a^3*b*(505*B - 209*C) + 15*a^2*b^2*(19*B - 121*C) + 10*a*b^3*(3*B - 11*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d) + (2*a*(14*b*B + 11*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*(81*a^2*B + 113*b^2*B + 209*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*(1145*a^2*b*B + 15*b^3*B + 539*a^3*C + 825*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a*d*Cos[c + d*x]^(5/2)) + (2*(675*a^4*B + 1025*a^2*b^2*B - 20*b^4*B + 1793*a^3*b*C + 55*a*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",9,7,44,0.1591,1,"{3029, 2989, 3047, 3055, 2998, 2816, 2994}"
920,1,571,0,1.7272225,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{\left(-15 a^2 C+18 a b B-16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-15 a^2 C+18 a b B+10 a b C-12 b^2 B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}+\frac{(a-b) \sqrt{a+b} \left(-15 a^2 C+18 a b B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \left(6 a^2 b B-5 a^3 C-4 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","-\frac{\left(-15 a^2 C+18 a b B-16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-15 a^2 C+18 a b B+10 a b C-12 b^2 B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}+\frac{(a-b) \sqrt{a+b} \left(-15 a^2 C+18 a b B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \left(6 a^2 b B-5 a^3 C-4 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"((a - b)*Sqrt[a + b]*(18*a*b*B - 15*a^2*C - 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d) - (Sqrt[a + b]*(18*a*b*B - 12*b^2*B - 15*a^2*C + 10*a*b*C - 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) - ((18*a*b*B - 15*a^2*C - 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) + ((6*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",9,9,44,0.2045,1,"{3029, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
921,1,479,0,1.1955996,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} \left(-3 a^2 C+4 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}","\frac{\sqrt{a+b} \left(-3 a^2 C+4 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"-((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (Sqrt[a + b]*(4*a*b*B - 3*a^2*C - 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",8,8,44,0.1818,1,"{3029, 2990, 3061, 3053, 2809, 2998, 2816, 2994}"
922,1,427,0,1.213424,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a C \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}","-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"-(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) - (Sqrt[a + b]*(2*b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (a*C*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,44,0.1818,1,"{3029, 3003, 3051, 2809, 2993, 2998, 2816, 2994}"
923,1,228,0,0.3925873,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)","A",4,4,44,0.09091,1,"{3029, 3006, 2809, 2816}"
924,1,230,0,0.4381107,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}","\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)","A",4,4,44,0.09091,1,"{3029, 2998, 2816, 2994}"
925,1,290,0,0.6291566,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} (a (B-3 C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 (a-b) \sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{a+b} (a (B-3 C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 (a-b) \sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(2*b*B + a*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))","A",5,5,44,0.1136,1,"{3029, 3000, 2998, 2816, 2994}"
926,1,363,0,0.9645145,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","-\frac{2 \sqrt{a+b} \left(a^2 (9 B-5 C)-2 a b (B+5 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B-10 a b C+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 (4 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 \sqrt{a+b} \left(a^2 (9 B-5 C)-2 a b (B+5 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B-10 a b C+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 (4 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2*B + 8*b^2*B - 10*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*b^2*B + a^2*(9*B - 5*C) - 2*a*b*(B + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))","A",6,6,44,0.1364,1,"{3029, 3000, 3055, 2998, 2816, 2994}"
927,1,620,0,1.8907277,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-5 a^2 C+4 a b B+b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}+\frac{\left(12 a^2 b B-15 a^3 C+7 a b^2 C-4 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^2 C+a b (12 B-5 C)+2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\left(12 a^2 b B-15 a^3 C+7 a b^2 C-4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}+\frac{\sqrt{a+b} \left(-15 a^2 C+12 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-5 a^2 C+4 a b B+b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}+\frac{\left(12 a^2 b B-15 a^3 C+7 a b^2 C-4 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^2 C+a b (12 B-5 C)+2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\left(12 a^2 b B-15 a^3 C+7 a b^2 C-4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}+\frac{\sqrt{a+b} \left(-15 a^2 C+12 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}",1,"-((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d) + ((a*b*(12*B - 5*C) - 15*a^2*C + 2*b^2*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a*b*B - 15*a^2*C - 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) + (2*a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((4*a*b*B - 5*a^2*C + b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)","A",9,9,44,0.2045,1,"{3029, 2989, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
928,1,500,0,1.4119024,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{(-3 a C+2 b B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{(-3 a C+2 b B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"((2*a*b*B - 3*a^2*C + b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) - ((2*b*B - 3*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a*b*B - 3*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",8,8,44,0.1818,1,"{3029, 2989, 3061, 3053, 2809, 2998, 2816, 2994}"
929,1,416,0,0.7256289,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"(-2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) + (2*(b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (2*a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",7,7,44,0.1591,1,"{3029, 2992, 2809, 2794, 2795, 2816, 2994}"
930,1,284,0,0.6402701,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{2 (B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{2 (B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"(2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) - (2*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,44,0.1136,1,"{3029, 2993, 2998, 2816, 2994}"
931,1,305,0,0.7445629,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 B+a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 (a (B-C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 B+a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 (a (B-C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}",1,"(2*(a^2*B - 2*b^2*B + a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(2*b*B + a*(B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,44,0.1136,1,"{3029, 3000, 2998, 2816, 2994}"
932,1,393,0,1.0804078,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \left(a^2 B+3 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(5 a^2 b B-3 a^3 C+6 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}+\frac{2 (a+2 b) (a (B-3 C)+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}","\frac{2 \left(a^2 B+3 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(5 a^2 b B-3 a^3 C+6 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}+\frac{2 (a+2 b) (a (B-3 C)+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}",1,"(-2*(5*a^2*b*B - 8*b^3*B - 3*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d) + (2*(a + 2*b)*(4*b*B + a*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*B - 4*b^2*B + 3*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",6,6,44,0.1364,1,"{3029, 3000, 3055, 2998, 2816, 2994}"
933,1,674,0,2.1242432,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(2 a^2 b B-5 a^3 C+9 a b^2 C-6 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(a^2 b (6 B-5 C)-15 a^3 C+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(2 a^2 b B-5 a^3 C+9 a b^2 C-6 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(a^2 b (6 B-5 C)-15 a^3 C+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) - ((a^2*b*(6*B - 5*C) - 3*b^3*(4*B - C) - 15*a^3*C + a*b^2*(2*B + 21*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(2*b*B - 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) + (2*a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])","A",9,9,44,0.2045,1,"{3029, 2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
934,1,545,0,1.4962554,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b C-3 a^3 C+a b^2 B+6 a b^2 C-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","-\frac{2 a \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b C-3 a^3 C+a b^2 B+6 a b^2 C-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(2*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*b^2*B - 3*b^3*B - 3*a^3*C - a^2*b*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",8,8,44,0.1818,1,"{3029, 2989, 3051, 2809, 2993, 2998, 2816, 2994}"
935,1,391,0,0.9893354,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a B+a C-b B-3 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d (a-b) (a+b)^{3/2}}","\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a B+a C-b B-3 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d (a-b) (a+b)^{3/2}}",1,"(-2*(3*a^2*B + b^2*B - 4*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*(3*a*B - b*B + a*C - 3*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^2*B + b^2*B - 4*a*b*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,44,0.1364,1,"{3029, 2999, 2993, 2998, 2816, 2994}"
936,1,429,0,1.1073781,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","-\frac{2 \left(6 a^2 b B-3 a^3 C-a b^2 C-2 b^3 B\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (B+C)+a b (3 B+C)+2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(6 a^2 b B-3 a^3 C-a b^2 C-2 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}","-\frac{2 \left(6 a^2 b B-3 a^3 C-a b^2 C-2 b^3 B\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (B+C)+a b (3 B+C)+2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(6 a^2 b B-3 a^3 C-a b^2 C-2 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*b^2*B - 3*a^2*(B + C) + a*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,44,0.1364,1,"{3029, 3000, 2993, 2998, 2816, 2994}"
937,1,456,0,1.2953503,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 b \left(8 a^2 b B-5 a^3 C+a b^2 C-4 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 B+C)-3 a^3 (B-C)+2 a b^2 (3 B-C)+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-15 a^2 b^2 B+6 a^3 b C+3 a^4 B-2 a b^3 C+8 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d (a-b) (a+b)^{3/2}}","\frac{2 b \left(8 a^2 b B-5 a^3 C+a b^2 C-4 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 B+C)-3 a^3 (B-C)+2 a b^2 (3 B-C)+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-15 a^2 b^2 B+6 a^3 b C+3 a^4 B-2 a b^3 C+8 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d (a-b) (a+b)^{3/2}}",1,"(2*(3*a^4*B - 15*a^2*b^2*B + 8*b^4*B + 6*a^3*b*C - 2*a*b^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d) + (2*(8*b^3*B - 3*a^3*(B - C) + 2*a*b^2*(3*B - C) - 3*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*b*B - 4*b^3*B - 5*a^3*C + a*b^2*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,44,0.1364,1,"{3029, 3000, 3055, 2998, 2816, 2994}"
938,1,156,0,0.2292857,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin ^3(c+d x) (5 a B+5 A b+4 b C)}{15 d}+\frac{\sin (c+d x) (5 a B+5 A b+4 b C)}{5 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a A+3 a C+3 b B)}{8 d}+\frac{1}{8} x (4 a A+3 a C+3 b B)+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}","-\frac{\sin ^3(c+d x) (5 a B+5 A b+4 b C)}{15 d}+\frac{\sin (c+d x) (5 a B+5 A b+4 b C)}{5 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a A+3 a C+3 b B)}{8 d}+\frac{1}{8} x (4 a A+3 a C+3 b B)+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"((4*a*A + 3*b*B + 3*a*C)*x)/8 + ((5*A*b + 5*a*B + 4*b*C)*Sin[c + d*x])/(5*d) + ((4*a*A + 3*b*B + 3*a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A*b + 5*a*B + 4*b*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{3033, 3023, 2748, 2635, 8, 2633}"
939,1,128,0,0.1385388,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac{1}{8} x (4 a B+4 A b+3 b C)+\frac{(a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}","\frac{\sin (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac{1}{8} x (4 a B+4 A b+3 b C)+\frac{(a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"((4*A*b + 4*a*B + 3*b*C)*x)/8 + ((3*a*A + 2*b*B + 2*a*C)*Sin[c + d*x])/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",3,3,37,0.08108,1,"{3033, 3023, 2734}"
940,1,113,0,0.0985468,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(a (3 b B-a C)+b^2 (3 A+2 C)\right)}{3 b d}+\frac{1}{2} x (a (2 A+C)+b B)+\frac{(3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}","\frac{\sin (c+d x) (a B+A b+b C)}{d}+\frac{1}{2} x (a (2 A+C)+b B)+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}-\frac{b C \sin ^3(c+d x)}{3 d}",1,"((b*B + a*(2*A + C))*x)/2 + ((b^2*(3*A + 2*C) + a*(3*b*B - a*C))*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)","A",2,2,31,0.06452,1,"{3023, 2734}"
941,1,69,0,0.1397303,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{1}{2} x (2 a B+2 A b+b C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}","\frac{1}{2} x (2 a B+2 A b+b C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*A*b + 2*a*B + b*C)*x)/2 + (a*A*ArcTanh[Sin[c + d*x]])/d + ((b*B + a*C)*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,37,0.1081,1,"{3033, 3023, 2735, 3770}"
942,1,52,0,0.1382503,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+x (a C+b B)+\frac{b C \sin (c+d x)}{d}","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+x (a C+b B)+\frac{b C \sin (c+d x)}{d}",1,"(b*B + a*C)*x + ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d","A",4,4,39,0.1026,1,"{3031, 3023, 2735, 3770}"
943,1,69,0,0.1680739,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{(a (A+2 C)+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+b C x","\frac{(a (A+2 C)+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+b C x",1,"b*C*x + ((2*b*B + a*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((A*b + a*B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,39,0.1026,1,"{3031, 3021, 2735, 3770}"
944,1,101,0,0.2201809,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{\tan (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"((A*b + a*B + 2*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*A + 3*b*B + 3*a*C)*Tan[c + d*x])/(3*d) + ((A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,39,0.1538,1,"{3031, 3021, 2748, 3767, 8, 3770}"
945,1,137,0,0.2391131,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) (2 a B+2 A b+3 b C)}{3 d}+\frac{(3 a A+4 a C+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a A+4 a C+4 b B)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}","\frac{\tan (c+d x) (2 a B+2 A b+3 b C)}{3 d}+\frac{(3 a A+4 a C+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a A+4 a C+4 b B)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((3*a*A + 4*b*B + 4*a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((2*A*b + 2*a*B + 3*b*C)*Tan[c + d*x])/(3*d) + ((3*a*A + 4*b*B + 4*a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,39,0.1795,1,"{3031, 3021, 2748, 3768, 3770, 3767, 8}"
946,1,165,0,0.255276,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac{\tan (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac{(3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}","\frac{\tan ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac{\tan (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac{(3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"((3*A*b + 3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A + 5*b*B + 5*a*C)*Tan[c + d*x])/(5*d) + ((3*A*b + 3*a*B + 4*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*a*A + 5*b*B + 5*a*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{3031, 3021, 2748, 3767, 3768, 3770}"
947,1,224,0,0.3250128,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right)}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)}{8 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)+\frac{b (2 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\sin (c+d x) \left(5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right)}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)}{8 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)+\frac{b (2 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*x)/8 + ((20*a*b*B + 5*a^2*(3*A + 2*C) + 2*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*A*b^2 + 10*a*b*B + 2*a^2*C + 4*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(5*b*B + 2*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d)","A",4,4,39,0.1026,1,"{3049, 3033, 3023, 2734}"
948,1,191,0,0.2260212,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right)+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}","\frac{\sin (c+d x) \left(4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right)+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"((8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x)/8 + ((4*a^2*b*B + 4*b^3*B - a^3*C + 4*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)","A",3,3,33,0.09091,1,"{3023, 2753, 2734}"
949,1,134,0,0.3317253,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\sin (c+d x) \left(2 a^2 C+6 a b B+3 A b^2+2 b^2 C\right)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b (2 A+C)+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{\sin (c+d x) \left(2 a^2 C+6 a b B+3 A b^2+2 b^2 C\right)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b (2 A+C)+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"((2*a^2*B + b^2*B + 2*a*b*(2*A + C))*x)/2 + (a^2*A*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 6*a*b*B + 2*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",5,5,39,0.1282,1,"{3049, 3033, 3023, 2735, 3770}"
950,1,126,0,0.3226062,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{1}{2} x \left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right)-\frac{b \sin (c+d x) (2 a A-2 a C-b B)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}","\frac{1}{2} x \left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right)-\frac{b \sin (c+d x) (2 a A-2 a C-b B)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*x)/2 + (a*(2*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a*A - b*B - 2*a*C)*Sin[c + d*x])/d - (b^2*(2*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d","A",5,5,41,0.1220,1,"{3047, 3033, 3023, 2735, 3770}"
951,1,118,0,0.3596957,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(a^2 (A+2 C)+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (a B+A b) \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+b x (2 a C+b B)-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}","\frac{\left(a^2 (A+2 C)+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (a B+A b) \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+b x (2 a C+b B)-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}",1,"b*(b*B + 2*a*C)*x + ((2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(A - 2*C)*Sin[c + d*x])/(2*d) + (a*(A*b + a*B)*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,41,0.1220,1,"{3047, 3031, 3023, 2735, 3770}"
952,1,141,0,0.3661207,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+2 A b^2\right)}{3 d}+\frac{\left(a^2 B+2 a b (A+2 C)+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (3 a B+2 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x","\frac{\tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+2 A b^2\right)}{3 d}+\frac{\left(a^2 B+2 a b (A+2 C)+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (3 a B+2 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x",1,"b^2*C*x + ((a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (a*(2*A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,41,0.1220,1,"{3047, 3031, 3021, 2735, 3770}"
953,1,184,0,0.4653756,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) \left(2 a^2 B+4 a A b+6 a b C+3 b^2 B\right)}{3 d}+\frac{\left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+2 A b^2\right)}{8 d}+\frac{a (2 a B+A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{\tan (c+d x) \left(2 a^2 B+4 a A b+6 a b C+3 b^2 B\right)}{3 d}+\frac{\left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+2 A b^2\right)}{8 d}+\frac{a (2 a B+A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"((8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B + 6*a*b*C)*Tan[c + d*x])/(3*d) + ((2*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A*b + 2*a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,41,0.1707,1,"{3047, 3031, 3021, 2748, 3767, 8, 3770}"
954,1,232,0,0.5070594,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(2 a^2 (4 A+5 C)+20 a b B+5 b^2 (2 A+3 C)\right)}{15 d}+\frac{\left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(a^2 (4 A+5 C)+10 a b B+2 A b^2\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)}{8 d}+\frac{a (5 a B+2 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\tan (c+d x) \left(2 a^2 (4 A+5 C)+20 a b B+5 b^2 (2 A+3 C)\right)}{15 d}+\frac{\left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(a^2 (4 A+5 C)+10 a b B+2 A b^2\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)}{8 d}+\frac{a (5 a B+2 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((20*a*b*B + 5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((2*A*b^2 + 10*a*b*B + a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(2*A*b + 5*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,41,0.1951,1,"{3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
955,1,327,0,0.6065918,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right)}{15 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left(6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right)}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(12 a^2 b B+a^3 C+3 a b^2 (5 A+4 C)+4 b^3 B\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right)}{16 d}+\frac{1}{16} x \left(6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right)+\frac{(a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}","\frac{\sin (c+d x) \left(5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right)}{15 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left(6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right)}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(12 a^2 b B+a^3 C+3 a b^2 (5 A+4 C)+4 b^3 B\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right)}{16 d}+\frac{1}{16} x \left(6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right)+\frac{(a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*x)/16 + ((30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A + 2*C) + 6*a*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((12*a^2*b*B + 4*b^3*B + a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d)","A",5,4,39,0.1026,1,"{3049, 3033, 3023, 2734}"
956,1,277,0,0.4192042,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(4 a^2 b^2 (20 A+13 C)+15 a^3 b B-3 a^4 C+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(30 a^2 b B-6 a^3 C+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 d}+\frac{1}{8} x \left(4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right)+\frac{\sin (c+d x) \left(3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right) (a+b \cos (c+d x))^2}{60 b d}+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}","\frac{\sin (c+d x) \left(4 a^2 b^2 (20 A+13 C)+15 a^3 b B-3 a^4 C+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(30 a^2 b B-6 a^3 C+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 d}+\frac{1}{8} x \left(4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right)+\frac{\sin (c+d x) \left(3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right) (a+b \cos (c+d x))^2}{60 b d}+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"((12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*x)/8 + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 4*b^4*(5*A + 4*C) + 4*a^2*b^2*(20*A + 13*C))*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + a*b^2*(100*A + 71*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((4*b^2*(5*A + 4*C) + 3*a*(5*b*B - a*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)","A",4,3,33,0.09091,1,"{3023, 2753, 2734}"
957,1,207,0,0.5635572,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\sin (c+d x) \left(16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{b \sin (c+d x) \cos (c+d x) \left(6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{1}{8} x \left(12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{\sin (c+d x) \left(16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{b \sin (c+d x) \cos (c+d x) \left(6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{1}{8} x \left(12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"((8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",6,5,39,0.1282,1,"{3049, 3033, 3023, 2735, 3770}"
958,1,192,0,0.5866823,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b \sin (c+d x) \left(a^2 (-(6 A-8 C))+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+\frac{1}{2} x \left(6 a^2 b B+2 a^3 C+3 a b^2 (2 A+C)+b^3 B\right)+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos (c+d x) (6 a A-5 a C-3 b B)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}","\frac{b \sin (c+d x) \left(a^2 (-(6 A-8 C))+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+\frac{1}{2} x \left(6 a^2 b B+2 a^3 C+3 a b^2 (2 A+C)+b^3 B\right)+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos (c+d x) (6 a A-5 a C-3 b B)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*x)/2 + (a^2*(3*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*b*B - a^2*(6*A - 8*C) + b^2*(3*A + 2*C))*Sin[c + d*x])/(3*d) - (b^2*(6*a*A - 3*b*B - 5*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d","A",6,6,41,0.1463,1,"{3047, 3049, 3033, 3023, 2735, 3770}"
959,1,204,0,0.645678,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b \sin (c+d x) \left(4 a^2 B+9 a A b-6 a b C-2 b^2 B\right)}{2 d}+\frac{a \left(a^2 (A+2 C)+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right)-\frac{b^2 \sin (c+d x) \cos (c+d x) (2 a B+4 A b-b C)}{2 d}+\frac{(2 a B+3 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}","-\frac{b \sin (c+d x) \left(4 a^2 B+9 a A b-6 a b C-2 b^2 B\right)}{2 d}+\frac{a \left(a^2 (A+2 C)+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right)-\frac{b^2 \sin (c+d x) \cos (c+d x) (2 a B+4 A b-b C)}{2 d}+\frac{(2 a B+3 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}",1,"(b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*x)/2 + (a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(9*a*A*b + 4*a^2*B - 2*b^2*B - 6*a*b*C)*Sin[c + d*x])/(2*d) - (b^2*(4*A*b + 2*a*B - b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + ((3*A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,41,0.1220,1,"{3047, 3033, 3023, 2735, 3770}"
960,1,196,0,0.6415917,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+3 A b^2\right)}{3 d}+\frac{\left(3 a^2 b (A+2 C)+a^3 B+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+b^2 x (3 a C+b B)","\frac{a \tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+3 A b^2\right)}{3 d}+\frac{\left(3 a^2 b (A+2 C)+a^3 B+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+b^2 x (3 a C+b B)",1,"b^2*(b*B + 3*a*C)*x + ((2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(5*A*b + 3*a*B - 6*b*C)*Sin[c + d*x])/(6*d) + (a*(3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + ((A*b + a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,5,41,0.1220,1,"{3047, 3031, 3023, 2735, 3770}"
961,1,223,0,0.6707894,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) \left(6 a^2 b (2 A+3 C)+4 a^3 B+16 a b^2 B+3 A b^3\right)}{6 d}+\frac{\left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left(3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right)}{24 d}+\frac{(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x","\frac{\tan (c+d x) \left(6 a^2 b (2 A+3 C)+4 a^3 B+16 a b^2 B+3 A b^3\right)}{6 d}+\frac{\left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left(3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right)}{24 d}+\frac{(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x",1,"b^3*C*x + ((12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*A*b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Tan[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,41,0.1220,1,"{3047, 3031, 3021, 2735, 3770}"
962,1,278,0,0.9179606,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(2 a^3 (4 A+5 C)+30 a^2 b B+15 a b^2 (2 A+3 C)+15 b^3 B\right)}{15 d}+\frac{\left(3 a^2 b (3 A+4 C)+3 a^3 B+12 a b^2 B+4 b^3 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(2 a^2 (4 A+5 C)+15 a b B+3 A b^2\right)}{30 d}+\frac{\tan (c+d x) \sec (c+d x) \left(15 a^2 b (3 A+4 C)+15 a^3 B+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{(5 a B+3 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}","\frac{\tan (c+d x) \left(2 a^3 (4 A+5 C)+30 a^2 b B+15 a b^2 (2 A+3 C)+15 b^3 B\right)}{15 d}+\frac{\left(3 a^2 b (3 A+4 C)+3 a^3 B+12 a b^2 B+4 b^3 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(2 a^2 (4 A+5 C)+15 a b B+3 A b^2\right)}{30 d}+\frac{\tan (c+d x) \sec (c+d x) \left(15 a^2 b (3 A+4 C)+15 a^3 B+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{(5 a B+3 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"((3*a^3*B + 12*a*b^2*B + 4*b^3*(A + 2*C) + 3*a^2*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((30*a^2*b*B + 15*b^3*B + 15*a*b^2*(2*A + 3*C) + 2*a^3*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((6*A*b^3 + 15*a^3*B + 50*a*b^2*B + 15*a^2*b*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a*(3*A*b^2 + 15*a*b*B + 2*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((3*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,7,41,0.1707,1,"{3047, 3031, 3021, 2748, 3767, 8, 3770}"
963,1,336,0,0.9133138,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{\tan (c+d x) \left(6 a^2 b (4 A+5 C)+8 a^3 B+30 a b^2 B+5 b^3 (2 A+3 C)\right)}{15 d}+\frac{\left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right)}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)}{16 d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}","\frac{\tan (c+d x) \left(6 a^2 b (4 A+5 C)+8 a^3 B+30 a b^2 B+5 b^3 (2 A+3 C)\right)}{15 d}+\frac{\left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right)}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)}{16 d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^3*B + 30*a*b^2*B + 5*b^3*(2*A + 3*C) + 6*a^2*b*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(6*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",9,8,41,0.1951,1,"{3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
964,1,445,0,1.0424077,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+280 a^3 b B+224 a b^3 B+8 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right) (a+b \cos (c+d x))^2}{70 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left(336 a^2 b B+24 a^3 C+4 a b^2 (126 A+103 C)+175 b^3 B\right)}{840 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(3 a^2 b^2 (63 A+50 C)+91 a^3 b B+4 a^4 C+112 a b^3 B+4 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)}{16 d}+\frac{1}{16} x \left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)+\frac{(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}","\frac{\sin (c+d x) \left(84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+280 a^3 b B+224 a b^3 B+8 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right) (a+b \cos (c+d x))^2}{70 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left(336 a^2 b B+24 a^3 C+4 a b^2 (126 A+103 C)+175 b^3 B\right)}{840 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(3 a^2 b^2 (63 A+50 C)+91 a^3 b B+4 a^4 C+112 a b^3 B+4 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)}{16 d}+\frac{1}{16} x \left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)+\frac{(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*x)/16 + ((280*a^3*b*B + 224*a*b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + ((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((91*a^3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^2*(126*A + 103*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(70*d) + ((7*b*B + 4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)","A",6,4,39,0.1026,1,"{3049, 3033, 3023, 2734}"
965,1,375,0,0.6826315,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\sin (c+d x) \left(24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \cos (c+d x))^2}{120 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 d}+\frac{1}{16} x \left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+24 a b^3 B+b^4 (6 A+5 C)\right)+\frac{\sin (c+d x) \left(4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right) (a+b \cos (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}","\frac{\sin (c+d x) \left(a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\sin (c+d x) \left(24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \cos (c+d x))^2}{120 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 d}+\frac{1}{16} x \left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+24 a b^3 B+b^4 (6 A+5 C)\right)+\frac{\sin (c+d x) \left(4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right) (a+b \cos (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}",1,"((32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*x)/16 + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*Sin[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)","A",5,3,33,0.09091,1,"{3023, 2753, 2734}"
966,1,290,0,0.9059554,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\sin (c+d x) \left(2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{\sin (c+d x) \left(12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right) (a+b \cos (c+d x))^2}{60 d}+\frac{b \sin (c+d x) \cos (c+d x) \left(130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+\frac{1}{8} x \left(16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}","\frac{\sin (c+d x) \left(2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{\sin (c+d x) \left(12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right) (a+b \cos (c+d x))^2}{60 d}+\frac{b \sin (c+d x) \cos (c+d x) \left(130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+\frac{1}{8} x \left(16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*x)/8 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(30*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",7,5,39,0.1282,1,"{3049, 3033, 3023, 2735, 3770}"
967,1,273,0,0.906428,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b \sin (c+d x) \left(a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{b^2 \sin (c+d x) \cos (c+d x) \left(a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right)+\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{12 d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}","\frac{b \sin (c+d x) \left(a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{b^2 \sin (c+d x) \cos (c+d x) \left(a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right)+\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{12 d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}",1,"((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x)/8 + (a^3*(4*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) + (b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d","A",7,6,41,0.1463,1,"{3047, 3049, 3033, 3023, 2735, 3770}"
968,1,274,0,0.9737557,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b \sin (c+d x) \left(a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{a^2 \left(a^2 (A+2 C)+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right)}{6 d}+\frac{1}{2} b x \left(12 a^2 b B+8 a^3 C+4 a b^2 (2 A+C)+b^3 B\right)-\frac{b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac{(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}","-\frac{b \sin (c+d x) \left(a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{a^2 \left(a^2 (A+2 C)+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right)}{6 d}+\frac{1}{2} b x \left(12 a^2 b B+8 a^3 C+4 a b^2 (2 A+C)+b^3 B\right)-\frac{b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac{(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}",1,"(b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*x)/2 + (a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(12*a^3*B - 24*a*b^2*B + a^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*A*b + a*B)*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,41,0.1463,1,"{3047, 3049, 3033, 3023, 2735, 3770}"
969,1,303,0,1.0770763,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{b \sin (c+d x) \left(4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right)}{6 d}+\frac{a \left(4 a^2 b (A+2 C)+a^3 B+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right)}{6 d}+\frac{\tan (c+d x) \left(a^2 (4 A+6 C)+15 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{6 d}+\frac{1}{2} b^2 x \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right)+\frac{(3 a B+4 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}","-\frac{b \sin (c+d x) \left(4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right)}{6 d}+\frac{a \left(4 a^2 b (A+2 C)+a^3 B+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right)}{6 d}+\frac{\tan (c+d x) \left(a^2 (4 A+6 C)+15 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{6 d}+\frac{1}{2} b^2 x \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right)+\frac{(3 a B+4 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}",1,"(b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*x)/2 + (a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(39*a^2*b*B - 6*b^3*B + 4*a*b^2*(11*A - 6*C) + 4*a^3*(2*A + 3*C))*Sin[c + d*x])/(6*d) - (b^2*(18*a*b*B + 3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) + ((12*A*b^2 + 15*a*b*B + a^2*(4*A + 6*C))*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + ((4*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,5,41,0.1220,1,"{3047, 3033, 3023, 2735, 3770}"
970,1,293,0,1.0648434,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{b^2 \sin (c+d x) \left(3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right)}{24 d}+\frac{a \tan (c+d x) \left(a^2 b (23 A+36 C)+8 a^3 B+36 a b^2 B+12 A b^3\right)}{12 d}+\frac{\left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+16 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+b^3 x (4 a C+b B)","-\frac{b^2 \sin (c+d x) \left(3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right)}{24 d}+\frac{a \tan (c+d x) \left(a^2 b (23 A+36 C)+8 a^3 B+36 a b^2 B+12 A b^3\right)}{12 d}+\frac{\left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+16 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+b^3 x (4 a C+b B)",1,"b^3*(b*B + 4*a*C)*x + ((8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*(32*a*b*B + 2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x])/(24*d) + (a*(12*A*b^3 + 8*a^3*B + 36*a*b^2*B + a^2*b*(23*A + 36*C))*Tan[c + d*x])/(12*d) + ((4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,5,41,0.1220,1,"{3047, 3031, 3023, 2735, 3770}"
971,1,314,0,1.0476624,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{\left(4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left(4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right)}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{60 d}+\frac{(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x","\frac{\tan (c+d x) \left(2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{\left(4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left(4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right)}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{60 d}+\frac{(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x",1,"b^4*C*x + ((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(30*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((12*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((4*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,5,41,0.1220,1,"{3047, 3031, 3021, 2735, 3770}"
972,1,381,0,1.3228853,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{\tan (c+d x) \left(8 a^3 b (4 A+5 C)+60 a^2 b^2 B+8 a^4 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{\left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+24 a^3 b B+32 a b^3 B+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(a^2 b (39 A+50 C)+16 a^3 B+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\tan (c+d x) \sec (c+d x) \left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+360 a^3 b B+336 a b^3 B+24 A b^4\right)}{240 d}+\frac{\tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{120 d}+\frac{(3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}","\frac{\tan (c+d x) \left(8 a^3 b (4 A+5 C)+60 a^2 b^2 B+8 a^4 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{\left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+24 a^3 b B+32 a b^3 B+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(a^2 b (39 A+50 C)+16 a^3 B+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\tan (c+d x) \sec (c+d x) \left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+360 a^3 b B+336 a b^3 B+24 A b^4\right)}{240 d}+\frac{\tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{120 d}+\frac{(3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}",1,"((24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^4*B + 60*a^2*b^2*B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*(4*A*b^3 + 16*a^3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((12*A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",9,7,41,0.1707,1,"{3047, 3031, 3021, 2748, 3767, 8, 3770}"
973,1,454,0,1.39733,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{\tan (c+d x) \left(84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)\right)}{105 d}+\frac{\left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left(a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right)}{840 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 d}+\frac{\tan (c+d x) \sec ^4(c+d x) \left(2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{70 d}+\frac{(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}","\frac{\tan (c+d x) \left(84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+224 a^3 b B+280 a b^3 B+35 b^4 (2 A+3 C)\right)}{105 d}+\frac{\left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left(a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right)}{840 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 d}+\frac{\tan (c+d x) \sec ^4(c+d x) \left(2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{70 d}+\frac{(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((224*a^3*b*B + 280*a*b^3*B + 35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[c + d*x])/(105*d) + ((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sec[c + d*x]^2*Tan[c + d*x])/(105*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Sec[c + d*x]^3*Tan[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(70*d) + ((4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)","A",10,8,41,0.1951,1,"{3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
974,1,256,0,0.551879,"\int (a+b \cos (c+d x))^3 \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^3*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{b \left(32 a^2 b^2 C+95 a^3 b B-83 a^4 C+80 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 d}+\frac{b \left(-23 a^2 C+35 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b^2 \left(130 a^2 b B-106 a^3 C+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(24 a^2 b^3 B-8 a^3 b^2 C+8 a^4 b B-8 a^5 C+9 a b^4 C+3 b^5 B\right)+\frac{b (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}","\frac{b \left(32 a^2 b^2 C+95 a^3 b B-83 a^4 C+80 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 d}+\frac{b \left(-23 a^2 C+35 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b^2 \left(130 a^2 b B-106 a^3 C+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(24 a^2 b^3 B-8 a^3 b^2 C+8 a^4 b B-8 a^5 C+9 a b^4 C+3 b^5 B\right)+\frac{b (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"((8*a^4*b*B + 24*a^2*b^3*B + 3*b^5*B - 8*a^5*C - 8*a^3*b^2*C + 9*a*b^4*C)*x)/8 + (b*(95*a^3*b*B + 80*a*b^3*B - 83*a^4*C + 32*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*d) + (b^2*(130*a^2*b*B + 45*b^3*B - 106*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(35*a*b*B - 23*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (b*(5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (b*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",5,3,48,0.06250,1,"{3015, 2753, 2734}"
975,1,176,0,0.3504119,"\int (a+b \cos (c+d x))^2 \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^2*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{b \left(16 a^2 b B-13 a^3 C+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(-14 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^3 b B-8 a^4 C+12 a b^3 B+3 b^4 C\right)+\frac{b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{b \left(16 a^2 b B-13 a^3 C+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(-14 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^3 b B-8 a^4 C+12 a b^3 B+3 b^4 C\right)+\frac{b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"((8*a^3*b*B + 12*a*b^3*B - 8*a^4*C + 3*b^4*C)*x)/8 + (b*(16*a^2*b*B + 4*b^3*B - 13*a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*d) + (b^2*(20*a*b*B - 14*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (b*(4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",4,3,48,0.06250,1,"{3015, 2753, 2734}"
976,1,120,0,0.2141505,"\int (a+b \cos (c+d x)) \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{2 b \left(-2 a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 b B-2 a^3 C+a b^2 C+b^3 B\right)+\frac{b^2 (3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{2 b \left(-2 a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 b B-2 a^3 C+a b^2 C+b^3 B\right)+\frac{b^2 (3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"((2*a^2*b*B + b^3*B - 2*a^3*C + a*b^2*C)*x)/2 + (2*b*(3*a*b*B - 2*a^2*C + b^2*C)*Sin[c + d*x])/(3*d) + (b^2*(3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",3,3,46,0.06522,1,"{3015, 2753, 2734}"
977,1,279,0,0.9353848,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\sin (c+d x) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+2 C)+2 b^3 B\right)}{3 b^4 d}-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2 (2 A+C)+8 a^3 b B-8 a^4 C+4 a b^3 B-b^4 (4 A+3 C)\right)}{8 b^5}+\frac{(b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}","\frac{\sin (c+d x) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+2 C)+2 b^3 B\right)}{3 b^4 d}-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2 (2 A+C)+8 a^3 b B-8 a^4 C+4 a b^3 B-b^4 (4 A+3 C)\right)}{8 b^5}+\frac{(b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"-((8*a^3*b*B + 4*a*b^3*B - 8*a^4*C - 4*a^2*b^2*(2*A + C) - b^4*(4*A + 3*C))*x)/(8*b^5) - (2*a^3*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) + ((3*a^2*b*B + 2*b^3*B - 3*a^3*C - a*b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*d) + ((4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",7,5,41,0.1220,1,"{3049, 3023, 2735, 2659, 205}"
978,1,206,0,0.5674193,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\sin (c+d x) \left(3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right)}{3 b^3 d}+\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{x \left(2 a^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right)}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}","\frac{\sin (c+d x) \left(3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right)}{3 b^3 d}+\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{x \left(2 a^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right)}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*(2*A + C))*x)/(2*b^4) + (2*a^2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",6,5,41,0.1220,1,"{3049, 3023, 2735, 2659, 205}"
979,1,142,0,0.3342233,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 a \left(A b^2-a (b B-a C)\right) \tan ^{-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{x \left(-\frac{2 a (b B-a C)}{b^2}+2 A+C\right)}{2 b}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{2 a \left(A b^2-a (b B-a C)\right) \tan ^{-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{x \left(b^2 (2 A+C)-2 a (b B-a C)\right)}{2 b^3}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"((2*A + C - (2*a*(b*B - a*C))/b^2)*x)/(2*b) - (2*a*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,39,0.1282,1,"{3049, 3023, 2735, 2659, 205}"
980,1,97,0,0.1473478,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"((b*B - a*C)*x)/b^2 + (2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)","A",4,4,33,0.1212,1,"{3023, 2735, 2659, 205}"
981,1,94,0,0.1370509,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}","-\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}",1,"(C*x)/b - (2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d) + (A*ArcTanh[Sin[c + d*x]])/(a*d)","A",4,4,39,0.1026,1,"{3057, 2659, 205, 3770}"
982,1,107,0,0.2621221,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}",1,"(2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^2*d) + (A*Tan[c + d*x])/(a*d)","A",5,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
983,1,154,0,0.5499499,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","-\frac{2 b \left(A b^2-a (b B-a C)\right) \tan ^{-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{\left(a^2 (A+2 C)-2 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(A b-a B) \tan (c+d x)}{a^2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 b \left(A b^2-a (b B-a C)\right) \tan ^{-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{\left(a^2 (A+2 C)-2 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(A b-a B) \tan (c+d x)}{a^2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(-2*b*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((A*b - a*B)*Tan[c + d*x])/(a^2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",6,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
984,1,214,0,0.8824087,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{\tan (c+d x) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}-\frac{\left(a^2 b (A+2 C)+a^3 (-B)-2 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(A b-a B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \tan ^{-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{\tan (c+d x) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}-\frac{\left(a^2 b (A+2 C)+a^3 (-B)-2 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(A b-a B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"(2*b^2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*A*b^3 - a^3*B - 2*a*b^2*B + a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",7,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
985,1,285,0,1.2819403,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + b*Cos[c + d*x]),x]","-\frac{2 b^3 \left(A b^2-a (b B-a C)\right) \tan ^{-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{\tan (c+d x) \left(a^2 b (2 A+3 C)-2 a^3 B-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{\left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)-4 a^3 b B-8 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 a^5 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)}{8 a^3 d}-\frac{(A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 a d}","-\frac{2 b^3 \left(A b^2-a (b B-a C)\right) \tan ^{-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{\tan (c+d x) \left(a^2 b (2 A+3 C)-2 a^3 B-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{\left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)-4 a^3 b B-8 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 a^5 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)}{8 a^3 d}-\frac{(A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 a d}",1,"(-2*b^3*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) + ((8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*a^5*d) - ((3*A*b^3 - 2*a^3*B - 3*a*b^2*B + a^2*b*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*d) + ((4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)","A",8,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
986,1,398,0,1.5970286,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\sin (c+d x) \left(-a^2 b^2 (6 A-7 C)+9 a^3 b B-12 a^4 C-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C-3 a^3 b B+4 a^4 C+4 a b^3 B-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C-3 a b B+3 A b^2-b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 b B-4 a^3 C-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{x \left(6 a^2 b B-8 a^3 C-2 a b^2 (2 A+C)+b^3 B\right)}{2 b^5}","-\frac{\sin (c+d x) \left(-a^2 b^2 (6 A-7 C)+9 a^3 b B-12 a^4 C-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C-3 a^3 b B+4 a^4 C+4 a b^3 B-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C-3 a b B+3 A b^2-b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 b B-4 a^3 C-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{x \left(6 a^2 b B-8 a^3 C-2 a b^2 (2 A+C)+b^3 B\right)}{2 b^5}",1,"((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*x)/(2*b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 2*a*b^2*(A - C) - 4*a^3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,41,0.1463,1,"{3047, 3049, 3023, 2735, 2659, 205}"
987,1,303,0,1.1173782,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sin (c+d x) \left(2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 C+3 a b^3 B-2 A b^4\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right)}{2 b^4}","\frac{\sin (c+d x) \left(2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 C+3 a b^3 B-2 A b^4\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right)}{2 b^4}",1,"((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - a*b^2*(A - 2*C) - 3*a^3*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,41,0.1463,1,"{3047, 3049, 3023, 2735, 2659, 205}"
988,1,168,0,0.461326,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(3 a^2 b^2 C+a^3 b B-2 a^4 C-2 a b^3 B+A b^4\right) \tan ^{-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 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}","-\frac{2 \left(3 a^2 b^2 C+a^3 b B-2 a^4 C-2 a b^3 B+A b^4\right) \tan ^{-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 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*x)/b^3 - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,39,0.1282,1,"{3031, 3023, 2735, 2659, 205}"
989,1,139,0,0.2262953,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(a^3 (-C)+a A b^2+2 a b^2 C-b^3 B\right) \tan ^{-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{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}","\frac{2 \left(a^3 (-C)+a A b^2+2 a b^2 C-b^3 B\right) \tan ^{-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{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*x)/b^2 + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,33,0.1212,1,"{3021, 2735, 2659, 205}"
990,1,147,0,0.3513644,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(2 a^2 A b+a^2 b C+a^3 (-B)-A b^3\right) \tan ^{-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{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}","-\frac{2 \left(2 a^2 A b+a^2 b C+a^3 (-B)-A b^3\right) \tan ^{-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{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"(-2*(2*a^2*A*b - A*b^3 - a^3*B + a^2*b*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,39,0.1282,1,"{3055, 3001, 3770, 2659, 205}"
991,1,211,0,0.7678247,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(3 a^2 A b^2-2 a^3 b B+a^4 C+a b^3 B-2 A b^4\right) \tan ^{-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{\tan (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{(2 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{2 \left(3 a^2 A b^2-2 a^3 b B+a^4 C+a b^3 B-2 A b^4\right) \tan ^{-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{\tan (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{(2 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((2*A*b^2 - a*b*B - a^2*(A - C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
992,1,307,0,1.402451,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 (A+2 C)-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 (A+2 C)-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(-2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
993,1,405,0,2.0036767,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C-4 a^3 b B+3 a^4 C+3 a b^3 B-4 A b^4\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+6 a^3 b B-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{\left(2 a^2 b (A+2 C)+a^3 (-B)-6 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{\tan (c+d x) \sec ^2(c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(-2 a^2 b (A-C)+a^3 B-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C-4 a^3 b B+3 a^4 C+3 a b^3 B-4 A b^4\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+6 a^3 b B-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{\left(2 a^2 b (A+2 C)+a^3 (-B)-6 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{\tan (c+d x) \sec ^2(c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(-2 a^2 b (A-C)+a^3 B-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
994,1,456,0,4.6368549,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sin (c+d x) \left(-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 C-12 a b^5 B+6 A b^6\right) \tan ^{-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{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a \left(2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right)+3 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{x \left(12 a^2 C-6 a b B+2 A b^2+b^2 C\right)}{2 b^5}","\frac{\sin (c+d x) \left(-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 C-12 a b^5 B+6 A b^6\right) \tan ^{-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{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a \left(2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right)+3 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{x \left(12 a^2 C-6 a b B+2 A b^2+b^2 C\right)}{2 b^5}",1,"((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*x)/(2*b^5) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 21*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C))*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,41,0.1463,1,"{3047, 3049, 3023, 2735, 2659, 205}"
995,1,314,0,2.8222732,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \sin (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4}","\frac{\sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \sin (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4}",1,"((b*B - 3*a*C)*x)/b^4 + ((2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,41,0.1463,1,"{3047, 3031, 3023, 2735, 2659, 205}"
996,1,233,0,0.7706187,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-3 a b^4 (A+2 C)+2 b^5 B\right) \tan ^{-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{\sin (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{C x}{b^3}","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-3 a b^4 (A+2 C)+2 b^5 B\right) \tan ^{-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{\sin (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{C x}{b^3}",1,"(C*x)/b^3 + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 3*a*b^4*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,39,0.1282,1,"{3031, 3021, 2735, 2659, 205}"
997,1,202,0,0.3592854,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(a^2 (-(2 A+C))+3 a b B-b^2 (A+2 C)\right) \tan ^{-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{\sin (c+d x) \left(a^2 b B+a^3 C-a b^2 (3 A+4 C)+2 b^3 B\right)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(a^2 (-(2 A+C))+3 a b B-b^2 (A+2 C)\right) \tan ^{-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{\sin (c+d x) \left(a^2 b B+a^3 C-a b^2 (3 A+4 C)+2 b^3 B\right)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-(((3*a*b*B - a^2*(2*A + C) - b^2*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{3021, 2754, 12, 2659, 205}"
998,1,238,0,0.9782074,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\left(5 a^2 A b^3-3 a^4 b (2 A+C)+a^3 b^2 B+2 a^5 B-2 A b^5\right) \tan ^{-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{\sin (c+d x) \left(-a^2 b^2 (5 A+2 C)+3 a^3 b B+a^4 (-C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{\left(5 a^2 A b^3-3 a^4 b (2 A+C)+a^3 b^2 B+2 a^5 B-2 A b^5\right) \tan ^{-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{\sin (c+d x) \left(-a^2 b^2 (5 A+2 C)+3 a^3 b B+a^4 (-C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"((5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((2*A*b^4 + 3*a^3*b*B - a^4*C - a^2*b^2*(5*A + 2*C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,5,39,0.1282,1,"{3055, 3001, 3770, 2659, 205}"
999,1,339,0,3.418581,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-5 a^3 b^3 B+6 a^5 b B-2 a^6 C+2 a b^5 B-6 A b^6\right) \tan ^{-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{\tan (c+d x) \left(11 a^2 A b^2+a^4 (-(2 A-3 C))-5 a^3 b B+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\tan (c+d x) \left(-a^2 b^2 (6 A+C)+4 a^3 b B-2 a^4 C-a b^3 B+3 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{(3 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-5 a^3 b^3 B+6 a^5 b B-2 a^6 C+2 a b^5 B-6 A b^6\right) \tan ^{-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{\tan (c+d x) \left(11 a^2 A b^2+a^4 (-(2 A-3 C))-5 a^3 b B+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\tan (c+d x) \left(-a^2 b^2 (6 A+C)+4 a^3 b B-2 a^4 C-a b^3 B+3 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{(3 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-(((15*a^2*A*b^4 - 6*A*b^6 + 6*a^5*b*B - 5*a^3*b^3*B + 2*a*b^5*B - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d)) - ((3*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((11*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 2*a*b^3*B - a^4*(2*A - 3*C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 + 4*a^3*b*B - a*b^3*B - 2*a^4*C - a^2*b^2*(6*A + C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
1000,1,462,0,5.1588963,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 C-6 a b^5 B+12 A b^6\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 (A+2 C)-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\tan (c+d x) \sec (c+d x) \left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\tan (c+d x) \sec (c+d x) \left(7 a^2 A b^2-5 a^3 b B+3 a^4 C+2 a b^3 B-4 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 C-6 a b^5 B+12 A b^6\right) \tan ^{-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{\tan (c+d x) \left(-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 (A+2 C)-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\tan (c+d x) \sec (c+d x) \left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\tan (c+d x) \sec (c+d x) \left(7 a^2 A b^2-5 a^3 b B+3 a^4 C+2 a b^3 B-4 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
1001,1,649,0,12.199242,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\sin (c+d x) \left(-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-68 a^4 b^3 B+65 a^2 b^5 B+24 a^6 b B-60 a^7 C-2 a b^6 (13 A-12 C)-6 b^7 B\right)}{6 b^5 d \left(a^2-b^2\right)^3}+\frac{a \left(-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-28 a^5 b^3 B+35 a^3 b^5 B+8 a^7 b B-20 a^8 C-20 a b^7 B+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}-\frac{\sin (c+d x) \cos ^4(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^3(c+d x) \left(a^2 b^2 (A+10 C)+2 a^3 b B-5 a^4 C-7 a b^3 B+4 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^3 b^3 B-8 a^5 b B+20 a^6 C-27 a b^5 B+12 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-a^4 b^2 (A-27 C)+a^2 b^4 (2 A-23 C)-11 a^3 b^3 B+4 a^5 b B-10 a^6 C+12 a b^5 B-b^6 (6 A-C)\right)}{2 b^4 d \left(a^2-b^2\right)^3}+\frac{x \left(20 a^2 C-8 a b B+2 A b^2+b^2 C\right)}{2 b^6}","\frac{\sin (c+d x) \left(-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-68 a^4 b^3 B+65 a^2 b^5 B+24 a^6 b B-60 a^7 C-2 a b^6 (13 A-12 C)-6 b^7 B\right)}{6 b^5 d \left(a^2-b^2\right)^3}+\frac{a \left(-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-28 a^5 b^3 B+35 a^3 b^5 B+8 a^7 b B-20 a^8 C-20 a b^7 B+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}-\frac{\sin (c+d x) \cos ^4(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^3(c+d x) \left(a^2 b^2 (A+10 C)+2 a^3 b B-5 a^4 C-7 a b^3 B+4 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^3 b^3 B-8 a^5 b B+20 a^6 C-27 a b^5 B+12 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-a^4 b^2 (A-27 C)+a^2 b^4 (2 A-23 C)-11 a^3 b^3 B+4 a^5 b B-10 a^6 C+12 a b^5 B-b^6 (6 A-C)\right)}{2 b^4 d \left(a^2-b^2\right)^3}+\frac{x \left(20 a^2 C-8 a b B+2 A b^2+b^2 C\right)}{2 b^6}",1,"((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) + (a*(8*A*b^8 + 8*a^7*b*B - 28*a^5*b^3*B + 35*a^3*b^5*B - 20*a*b^7*B - a^6*b^2*(2*A - 69*C) + 7*a^4*b^4*(A - 12*C) - 8*a^2*b^6*(A - 5*C) - 20*a^8*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((24*a^6*b*B - 68*a^4*b^3*B + 65*a^2*b^5*B - 6*b^7*B - a^5*b^2*(6*A - 167*C) + a^3*b^4*(17*A - 146*C) - 2*a*b^6*(13*A - 12*C) - 60*a^7*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) - ((4*a^5*b*B - 11*a^3*b^3*B + 12*a*b^5*B - a^4*b^2*(A - 27*C) + a^2*b^4*(2*A - 23*C) - b^6*(6*A - C) - 10*a^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((4*A*b^4 + 2*a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,6,41,0.1463,1,"{3047, 3049, 3023, 2735, 2659, 205}"
1002,1,461,0,9.7605553,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{\sin (c+d x) \left(23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right) \tan ^{-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{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left(3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x (b B-4 a C)}{b^5}","-\frac{\sin (c+d x) \left(23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right) \tan ^{-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{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left(3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x (b B-4 a C)}{b^5}",1,"((b*B - 4*a*C)*x)/b^5 - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,6,41,0.1463,1,"{3047, 3031, 3023, 2735, 2659, 205}"
1003,1,349,0,2.3935531,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-4 a b^6 (A+2 C)+2 b^7 B\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\sin (c+d x) \left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 C-16 a b^5 B+4 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \sin (c+d x) \left(a^2 b^2 (3 A+8 C)-3 a^4 C-5 a b^3 B+2 A b^4\right)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{C x}{b^4}","-\frac{\left(-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-4 a b^6 (A+2 C)+2 b^7 B\right) \tan ^{-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{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\sin (c+d x) \left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 C-16 a b^5 B+4 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \sin (c+d x) \left(a^2 b^2 (3 A+8 C)-3 a^4 C-5 a b^3 B+2 A b^4\right)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{C x}{b^4}",1,"(C*x)/b^4 - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((4*A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,41,0.1463,1,"{3047, 3031, 3021, 2735, 2659, 205}"
1004,1,314,0,0.9187847,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\left(-a^2 b (4 A+3 C)+a^3 B+4 a b^2 B-b^3 (A+2 C)\right) \tan ^{-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{\sin (c+d x) \left(a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a^4 b B+2 a^5 C+a b^4 (13 A+18 C)-6 b^5 B\right)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{\left(-a^2 b (4 A+3 C)+a^3 B+4 a b^2 B-b^3 (A+2 C)\right) \tan ^{-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{\sin (c+d x) \left(a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a^4 b B+2 a^5 C+a b^4 (13 A+18 C)-6 b^5 B\right)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + a^3*b^2*(2*A - 5*C) + 2*a^5*C + a*b^4*(13*A + 18*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,39,0.1538,1,"{3031, 3021, 2754, 12, 2659, 205}"
1005,1,299,0,0.7752737,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(a^3 (-(2 A+C))+4 a^2 b B-a b^2 (3 A+4 C)+b^3 B\right) \tan ^{-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{\sin (c+d x) \left(-a^2 b^2 (11 A+10 C)+2 a^3 b B+a^4 C+13 a b^3 B-2 b^4 (2 A+3 C)\right)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(2 a^2 b B+a^3 C-a b^2 (5 A+6 C)+3 b^3 B\right)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","-\frac{\left(a^3 (-(2 A+C))+4 a^2 b B-a b^2 (3 A+4 C)+b^3 B\right) \tan ^{-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{\sin (c+d x) \left(-a^2 b^2 (11 A+10 C)+2 a^3 b B+a^4 C+13 a b^3 B-2 b^4 (2 A+3 C)\right)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(2 a^2 b B+a^3 C-a b^2 (5 A+6 C)+3 b^3 B\right)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"-(((4*a^2*b*B + b^3*B - a^3*(2*A + C) - a*b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*b*B + 3*b^3*B + a^3*C - a*b^2*(5*A + 6*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*b*B + 13*a*b^3*B + a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{3021, 2754, 12, 2659, 205}"
1006,1,345,0,2.5504484,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \tan ^{-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{\sin (c+d x) \left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 C+3 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{\left(-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \tan ^{-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{\sin (c+d x) \left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 C+3 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-(((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,5,39,0.1282,1,"{3055, 3001, 3770, 2659, 205}"
1007,1,480,0,10.5378145,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-2 a b^7 B+8 A b^8\right) \tan ^{-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{\tan (c+d x) \left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\tan (c+d x) \left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+a b^5 B-4 A b^6\right)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \left(-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 C-a b^3 B+4 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}","-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-2 a b^7 B+8 A b^8\right) \tan ^{-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{\tan (c+d x) \left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\tan (c+d x) \left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+a b^5 B-4 A b^6\right)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \left(-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 C-a b^3 B+4 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}",1,"-(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - ((4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
1008,1,657,0,12.8585691,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","\frac{b \left(-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{\tan (c+d x) \left(a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 (A+2 C)-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\tan (c+d x) \sec (c+d x) \left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\tan (c+d x) \sec (c+d x) \left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left(-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{b \left(-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{\tan (c+d x) \left(a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 (A+2 C)-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\tan (c+d x) \sec (c+d x) \left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\tan (c+d x) \sec (c+d x) \left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left(-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"(b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",9,5,41,0.1220,1,"{3055, 3001, 3770, 2659, 205}"
1009,1,23,0,0.0231874,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","x (b B-a C)+\frac{b C \sin (c+d x)}{d}","x (b B-a C)+\frac{b C \sin (c+d x)}{d}",1,"(b*B - a*C)*x + (b*C*Sin[c + d*x])/d","A",3,2,48,0.04167,1,"{24, 2637}"
1010,1,61,0,0.1157176,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 (b B-2 a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}+C x","\frac{2 (b B-2 a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}+C x",1,"C*x + (2*(b*B - 2*a*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",4,4,48,0.08333,1,"{24, 2735, 2659, 205}"
1011,1,110,0,0.1810533,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{2 \left(a^2 (-C)+a b B-b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 \left(a^2 (-C)+a b B-b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*(a*b*B - a^2*C - b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,48,0.1042,1,"{24, 2754, 12, 2659, 205}"
1012,1,175,0,0.4059258,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\left(2 a^2 b B-2 a^3 C-4 a b^2 C+b^3 B\right) \tan ^{-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{b \left(-4 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b (b B-2 a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(2 a^2 b B-2 a^3 C-4 a b^2 C+b^3 B\right) \tan ^{-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{b \left(-4 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b (b B-2 a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((2*a^2*b*B + b^3*B - 2*a^3*C - 4*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(3*a*b*B - 4*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,5,48,0.1042,1,"{24, 2754, 12, 2659, 205}"
1013,1,249,0,0.7462192,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^5} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^5,x]","\frac{\left(-7 a^2 b^2 C+2 a^3 b B-2 a^4 C+3 a b^3 B-b^4 C\right) \tan ^{-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{b \left(11 a^2 b B-13 a^3 C-17 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{b \left(-7 a^2 C+5 a b B-3 b^2 C\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{\left(-7 a^2 b^2 C+2 a^3 b B-2 a^4 C+3 a b^3 B-b^4 C\right) \tan ^{-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{b \left(11 a^2 b B-13 a^3 C-17 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{b \left(-7 a^2 C+5 a b B-3 b^2 C\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((2*a^3*b*B + 3*a*b^3*B - 2*a^4*C - 7*a^2*b^2*C - b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (b*(5*a*b*B - 7*a^2*C - 3*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2*b*B + 4*b^3*B - 13*a^3*C - 17*a*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,5,48,0.1042,1,"{24, 2754, 12, 2659, 205}"
1014,1,416,0,0.9306381,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \sin (c+d x) \left(24 a^2 b B-16 a^3 C-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(24 a^2 b B-16 a^3 C-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-6 a^2 b^2 (7 A+4 C)+24 a^3 b B-16 a^4 C+57 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}","\frac{2 \sin (c+d x) \left(24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \sin (c+d x) \left(24 a^2 b B-16 a^3 C-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(24 a^2 b B-16 a^3 C-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-6 a^2 b^2 (7 A+4 C)+24 a^3 b B-16 a^4 C+57 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}",1,"(2*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 6*a^2*b^2*(7*A + 4*C) + 21*b^4*(9*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*b*B + 75*b^3*B - 16*a^3*C - 6*a*b^2*(7*A + 6*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2*b*B + 75*b^3*B - 16*a^3*C - 6*a*b^2*(7*A + 6*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(63*A*b^2 - 36*a*b*B + 24*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) + (2*(3*b*B - 2*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)","A",9,8,43,0.1860,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1015,1,321,0,0.5742114,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 b B-8 a^3 C-a b^2 (35 A+19 C)-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}","\frac{2 \sin (c+d x) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 b B-8 a^3 C-a b^2 (35 A+19 C)-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(-2*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - a*b^2*(35*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)","A",8,8,41,0.1951,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1016,1,237,0,0.3696535,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a (5 b B-2 a C)+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a (5 b B-2 a C)+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*(3*b^2*(5*A + 3*C) + a*(5*b*B - 2*a*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,35,0.2000,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1017,1,240,0,0.692933,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \left(3 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{2 \left(3 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - (a^2 - b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,41,0.2195,1,"{3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1018,1,217,0,0.6660053,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{(a A+2 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}","\frac{(a A+2 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"-(((A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a*A + 2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",9,9,43,0.2093,1,"{3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1019,1,299,0,1.0572032,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{\left(-4 a^2 (A+2 C)-4 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+3 A b+8 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{(4 a B+A b) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\left(-4 a^2 (A+2 C)-4 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+3 A b+8 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{(4 a B+A b) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((3*A*b + 4*a*B + 8*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^2 - 4*a*b*B - 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1020,1,399,0,1.5296841,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\tan (c+d x) \left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(-8 a^2 (2 A+3 C)-18 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 b (A+2 C)+8 a^3 B-2 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","-\frac{\tan (c+d x) \left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(-8 a^2 (2 A+3 C)-18 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 b (A+2 C)+8 a^3 B-2 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b^2 - 18*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b^3 + 8*a^3*B - 2*a*b^2*B + 4*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1021,1,518,0,1.2873295,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \sin (c+d x) \left(88 a^2 b B-48 a^3 C-6 a b^2 (33 A+34 C)+539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \sin (c+d x) \left(-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}","\frac{2 \sin (c+d x) \left(24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \sin (c+d x) \left(88 a^2 b B-48 a^3 C-6 a b^2 (33 A+34 C)+539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \sin (c+d x) \left(-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-18 a^2 b^2 (11 A+8 C)+88 a^3 b B-48 a^4 C+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}",1,"(2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)","A",10,8,43,0.1860,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1022,1,408,0,0.8202514,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \sin (c+d x) \left(18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 C-246 a b^3 B-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}","\frac{2 \sin (c+d x) \left(8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \sin (c+d x) \left(18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 C-246 a b^3 B-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(-2*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)","A",9,8,41,0.1951,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1023,1,315,0,0.5187536,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 b B-6 a^3 C+2 a b^2 (70 A+41 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}","\frac{2 \sin (c+d x) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 b B-6 a^3 C+2 a b^2 (70 A+41 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}",1,"(2*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,7,35,0.2000,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1024,1,306,0,1.0141129,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{2 \left(5 a^2 b B+3 a^3 C-3 a b^2 (5 A+C)-5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","-\frac{2 \left(5 a^2 b B+3 a^3 C-3 a b^2 (5 A+C)-5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*b*B - 5*b^3*B + 3*a^3*C - 3*a*b^2*(5*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",10,9,41,0.2195,1,"{3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1025,1,286,0,1.0343511,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\left(a^2 (3 A-2 C)+6 a b B+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 a A-8 a C-6 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a B+3 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}","\frac{\left(a^2 (3 A-2 C)+6 a b B+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 a A-8 a C-6 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a B+3 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"-((3*a*A - 6*b*B - 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((6*a*b*B + a^2*(3*A - 2*C) + 2*b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",10,10,43,0.2326,1,"{3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1026,1,307,0,1.0757933,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(4 a^2 B+a b (7 A+8 C)+8 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (A+2 C)+12 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a B+5 A b-8 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{(4 a B+3 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}","\frac{\left(4 a^2 B+a b (7 A+8 C)+8 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (A+2 C)+12 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a B+5 A b-8 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{(4 a B+3 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"-((5*A*b + 4*a*B - 8*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((4*a^2*B + 8*b^2*B + a*b*(7*A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 12*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,9,43,0.2093,1,"{3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1027,1,399,0,1.549936,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+42 a b B+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-12 a^2 b (A+2 C)-8 a^3 B-6 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+42 a b B+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-12 a^2 b (A+2 C)-8 a^3 B-6 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"-((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((42*a*b*B + 8*a^2*(2*A + 3*C) + b^2*(17*A + 48*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^3 - 8*a^3*B - 6*a*b^2*B - 12*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((A*b + 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1028,1,503,0,2.0425714,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{\tan (c+d x) \left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a^2 d}-\frac{\left(-12 a^2 b (19 A+28 C)-128 a^3 B-136 a b^2 B+3 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec (c+d x) \left(12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{96 a d}+\frac{(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}","-\frac{\tan (c+d x) \left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a^2 d}-\frac{\left(-12 a^2 b (19 A+28 C)-128 a^3 B-136 a b^2 B+3 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec (c+d x) \left(12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{96 a d}+\frac{(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(192*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((3*A*b^3 - 128*a^3*B - 136*a*b^2*B - 12*a^2*b*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1029,1,629,0,1.514512,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac{2 \sin (c+d x) \left(104 a^2 b B-48 a^3 C-2 a b^2 (143 A+166 C)+1053 b^3 B\right) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{2 \sin (c+d x) \left(-10 a^2 b^2 (143 A+124 C)+520 a^3 b B-240 a^4 C+4355 a b^3 B+539 b^4 (13 A+11 C)\right) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac{2 \sin (c+d x) \left(-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)+3315 a^3 b^3 B+520 a^5 b B-240 a^6 C+48165 a b^5 B+1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}","\frac{2 \sin (c+d x) \left(24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac{2 \sin (c+d x) \left(104 a^2 b B-48 a^3 C-2 a b^2 (143 A+166 C)+1053 b^3 B\right) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{2 \sin (c+d x) \left(-10 a^2 b^2 (143 A+124 C)+520 a^3 b B-240 a^4 C+4355 a b^3 B+539 b^4 (13 A+11 C)\right) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac{2 \sin (c+d x) \left(-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+520 a^4 b B-240 a^5 C+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)+3315 a^3 b^3 B+520 a^5 b B-240 a^6 C+48165 a b^5 B+1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}",1,"(2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) + (2*(104*a^2*b*B + 1053*b^3*B - 48*a^3*C - 2*a*b^2*(143*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 121*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) + (2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d)","A",11,8,43,0.1860,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1030,1,510,0,1.0964254,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \sin (c+d x) \left(110 a^2 b B-40 a^3 C-5 a b^2 (99 A+67 C)-539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \sin (c+d x) \left(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 C-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 C-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B+110 a^4 b B-40 a^5 C-15 a b^4 (319 A+247 C)-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}","\frac{2 \sin (c+d x) \left(8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \sin (c+d x) \left(110 a^2 b B-40 a^3 C-5 a b^2 (99 A+67 C)-539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \sin (c+d x) \left(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 C-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 C-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B+110 a^4 b B-40 a^5 C-15 a b^4 (319 A+247 C)-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(-2*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 15*a^3*b^2*(33*A + 17*C) - 15*a*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 5*a*b^2*(99*A + 67*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 - 22*a*b*B + 8*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)","A",10,8,41,0.1951,1,"{3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1031,1,402,0,0.7557974,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sin (c+d x) \left(-10 a^2 C+45 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \sin (c+d x) \left(45 a^2 b B-10 a^3 C+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(45 a^2 b B-10 a^3 C+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 b^2 (161 A+93 C)+45 a^3 b B-10 a^4 C+435 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}","\frac{2 \sin (c+d x) \left(-10 a^2 C+45 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \sin (c+d x) \left(45 a^2 b B-10 a^3 C+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(45 a^2 b B-10 a^3 C+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 b^2 (161 A+93 C)+45 a^3 b B-10 a^4 C+435 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}",1,"(2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(161*A + 93*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)","A",9,7,35,0.2000,1,"{3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1032,1,383,0,1.3780957,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \sin (c+d x) \left(15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-10 a^2 b^2 (7 A-C)+56 a^3 b B+15 a^4 C-56 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B+15 a^3 C+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}","\frac{2 \sin (c+d x) \left(15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-10 a^2 b^2 (7 A-C)+56 a^3 b B+15 a^4 C-56 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B+15 a^3 C+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*b*B - 56*a*b^3*B - 10*a^2*b^2*(7*A - C) + 15*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",11,9,41,0.2195,1,"{3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1033,1,357,0,1.4006889,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\left(a^3 (15 A-16 C)+20 a^2 b B+4 a b^2 (15 A+4 C)+10 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(a^2 (-(15 A-46 C))+70 a b B+6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a B+5 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (15 a A-16 a C-10 b B) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}","\frac{\left(a^3 (15 A-16 C)+20 a^2 b B+4 a b^2 (15 A+4 C)+10 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(a^2 (-(15 A-46 C))+70 a b B+6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a B+5 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (15 a A-16 a C-10 b B) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}",1,"((70*a*b*B - a^2*(15*A - 46*C) + 6*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((20*a^2*b*B + 10*b^3*B + a^3*(15*A - 16*C) + 4*a*b^2*(15*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*(5*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(15*a*A - 10*b*B - 16*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d","A",11,10,43,0.2326,1,"{3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1034,1,372,0,1.4475861,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\left(a^2 b (33 A+16 C)+12 a^3 B+48 a b^2 B+8 b^3 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(12 a^2 B+a b (27 A-56 C)-24 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 (A+2 C)+20 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}","\frac{\left(a^2 b (33 A+16 C)+12 a^3 B+48 a b^2 B+8 b^3 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(12 a^2 B+a b (27 A-56 C)-24 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 (A+2 C)+20 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}",1,"-((12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((12*a^3*B + 48*a*b^2*B + 8*b^3*(3*A + C) + a^2*b*(33*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(21*A*b + 12*a*B - 8*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + ((5*A*b + 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,10,43,0.2326,1,"{3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1035,1,407,0,1.5709368,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 b (A+2 C)+8 a^3 B+30 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 b (A+2 C)+8 a^3 B+30 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}",1,"-((54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((66*a^2*b*B + 48*b^3*B + 8*a^3*(2*A + 3*C) + a*b^2*(59*A + 96*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 20*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + ((5*A*b + 6*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,9,43,0.2093,1,"{3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1036,1,502,0,2.109981,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) \left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(4 a^2 b (89 A+132 C)+128 a^3 B+472 a b^2 B+b^3 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-40 a b^3 B+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{(8 a B+5 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}","\frac{\tan (c+d x) \left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(4 a^2 b (89 A+132 C)+128 a^3 B+472 a b^2 B+b^3 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-40 a b^3 B+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{(8 a B+5 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}",1,"-((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((128*a^3*B + 472*a*b^2*B + 4*a^2*b*(89*A + 132*C) + b^3*(133*A + 384*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^4 - 160*a^3*b*B - 40*a*b^3*B - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((5*A*b^2 + 24*a*b*B + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + ((5*A*b + 8*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1037,1,624,0,2.7102743,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{\tan (c+d x) \left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)}}{1920 a^2 d}-\frac{\left(-4 a^2 b^2 (809 A+1180 C)-256 a^4 (4 A+5 C)-3560 a^3 b B-1330 a b^3 B+15 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{128 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{240 d}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{960 a d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}","-\frac{\tan (c+d x) \left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)}}{1920 a^2 d}-\frac{\left(-4 a^2 b^2 (809 A+1180 C)-256 a^4 (4 A+5 C)-3560 a^3 b B-1330 a b^3 B+15 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{128 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{240 d}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{960 a d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}",1,"((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(1920*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((15*A*b^4 - 3560*a^3*b*B - 1330*a*b^3*B - 256*a^4*(4*A + 5*C) - 4*a^2*b^2*(809*A + 1180*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(1920*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(128*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(1920*a^2*d) + ((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + ((A*b + 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",13,10,43,0.2326,1,"{3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1038,1,285,0,0.6555004,"\int (a+b \cos (c+d x))^{3/2} \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{2 b \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B-146 a^3 C+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}","\frac{2 b \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 b B-146 a^3 C+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*(161*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",9,7,50,0.1400,1,"{3015, 2753, 2752, 2663, 2661, 2655, 2653}"
1039,1,221,0,0.4938574,"\int \sqrt{a+b \cos (c+d x)} \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-17 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-17 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(20*a*b*B - 17*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",8,7,50,0.1400,1,"{3015, 2753, 2752, 2663, 2661, 2655, 2653}"
1040,1,344,0,0.7146052,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \left(24 a^2 C-28 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}-\frac{2 \left(-2 a^2 b^2 (35 A+16 C)+56 a^3 b B-48 a^4 C+49 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(56 a^2 b B-48 a^3 C-2 a b^2 (35 A+22 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-6 a C) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}","\frac{2 \sin (c+d x) \left(24 a^2 C-28 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}-\frac{2 \left(-2 a^2 b^2 (35 A+16 C)+56 a^3 b B-48 a^4 C+49 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(56 a^2 b B-48 a^3 C-2 a b^2 (35 A+22 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-6 a C) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}",1,"(2*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 2*a*b^2*(35*A + 22*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*b*B + 49*a*b^3*B - 48*a^4*C - 5*b^4*(7*A + 5*C) - 2*a^2*b^2*(35*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)","A",8,7,43,0.1628,1,"{3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1041,1,258,0,0.423549,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(10 a^2 b B-8 a^3 C-a b^2 (15 A+7 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}","\frac{2 \left(10 a^2 b B-8 a^3 C-a b^2 (15 A+7 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - a*b^2*(15*A + 7*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)","A",7,7,41,0.1707,1,"{3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1042,1,188,0,0.2345956,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","\frac{2 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,35,0.1714,1,"{3023, 2752, 2663, 2661, 2655, 2653}"
1043,1,189,0,0.4491828,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,41,0.1951,1,"{3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1044,1,220,0,0.6745121,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{(A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{(A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"-((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)","A",9,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1045,1,303,0,1.0197294,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(4 a^2 (A+2 C)-4 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 A b-4 a B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 A b-4 a B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}","\frac{\left(4 a^2 (A+2 C)-4 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 A b-4 a B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 A b-4 a B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 - 4*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",10,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1046,1,405,0,1.4857084,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}+\frac{\left(8 a^2 (2 A+3 C)-6 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(4 a^2 b (A+2 C)-8 a^3 B-6 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-6 a B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}+\frac{\left(8 a^2 (2 A+3 C)-6 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(4 a^2 b (A+2 C)-8 a^3 B-6 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-6 a B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"-((15*A*b^2 - 18*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((5*A*b^2 - 6*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^3 - 8*a^3*B - 6*a*b^2*B + 4*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 - 18*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^3*d) - ((5*A*b - 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",11,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1047,1,426,0,0.9223575,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \cos (c+d x) \left(6 a^2 C-5 a b B+5 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \left(20 a^2 b B-24 a^3 C-3 a b^2 (5 A-3 C)-5 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(40 a^2 b B-48 a^3 C-6 a b^2 (5 A+2 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 a^4 C-25 a b^3 B+3 b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \cos (c+d x) \left(6 a^2 C-5 a b B+5 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \left(20 a^2 b B-24 a^3 C-3 a b^2 (5 A-3 C)-5 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(40 a^2 b B-48 a^3 C-6 a b^2 (5 A+2 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 a^4 C-25 a b^3 B+3 b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(40*a^3*b*B - 25*a*b^3*B - 6*a^2*b^2*(5*A - 4*C) - 48*a^4*C + 3*b^4*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 6*a*b^2*(5*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 3*a*b^2*(5*A - 3*C) - 24*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",8,8,43,0.1860,1,"{3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1048,1,280,0,0.5160791,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C-6 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 b B-8 a^3 C-a b^2 (3 A-5 C)-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}","\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C-6 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 b B-8 a^3 C-a b^2 (3 A-5 C)-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(2*(6*a^2*b*B - 3*b^3*B - a*b^2*(3*A - 5*C) - 8*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - 6*a*b*B + 8*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)","A",7,7,41,0.1707,1,"{3031, 3023, 2752, 2663, 2661, 2655, 2653}"
1049,1,219,0,0.2894157,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{3021, 2752, 2663, 2661, 2655, 2653}"
1050,1,271,0,0.7872129,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",9,9,41,0.2195,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1051,1,313,0,1.0743674,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{b \sin (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}","-\frac{b \sin (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])","A",10,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1052,1,416,0,1.5860038,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b \sin (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-12 a b^2 B+15 A b^3\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-a^2 (7 A b-8 b C)+4 a^3 B-12 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)-12 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-4 a B) \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}","\frac{b \sin (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-12 a b^2 B+15 A b^3\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-a^2 (7 A b-8 b C)+4 a^3 B-12 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)-12 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-4 a B) \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}",1,"-((15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 - 12*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])","A",11,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1053,1,622,0,1.6749868,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(-2 a^2 b^2 (A-6 C)+5 a^3 b B-8 a^4 C-9 a b^3 B+6 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (15 A-71 C)+30 a^3 b B-48 a^4 C-50 a b^3 B+b^4 (35 A-3 C)\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \left(-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+40 a^4 b B-64 a^5 C+2 a b^4 (20 A-7 C)+5 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}+\frac{2 \left(-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+80 a^4 b B-128 a^5 C+a b^4 (45 A+17 C)-5 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-140 a^3 b^3 B+80 a^5 b B-128 a^6 C+40 a b^5 B-3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(-2 a^2 b^2 (A-6 C)+5 a^3 b B-8 a^4 C-9 a b^3 B+6 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (15 A-71 C)+30 a^3 b B-48 a^4 C-50 a b^3 B+b^4 (35 A-3 C)\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \left(-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+40 a^4 b B-64 a^5 C+2 a b^4 (20 A-7 C)+5 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}+\frac{2 \left(-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+80 a^4 b B-128 a^5 C+a b^4 (45 A+17 C)-5 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-140 a^3 b^3 B+80 a^5 b B-128 a^6 C+40 a b^5 B-3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(80*a^5*b*B - 140*a^3*b^3*B + 40*a*b^5*B - 4*a^4*b^2*(10*A - 53*C) + 5*a^2*b^4*(15*A - 11*C) - 128*a^6*C - 3*b^6*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(80*a^4*b*B - 80*a^2*b^3*B - 5*b^5*B - 4*a^3*b^2*(10*A - 29*C) - 128*a^5*C + a*b^4*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(6*A*b^4 + 5*a^3*b*B - 9*a*b^3*B - 2*a^2*b^2*(A - 6*C) - 8*a^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(40*a^4*b*B - 65*a^2*b^3*B + 5*b^5*B - 2*a^3*b^2*(10*A - 49*C) + 2*a*b^4*(20*A - 7*C) - 64*a^5*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*b*B - 50*a*b^3*B - a^2*b^2*(15*A - 71*C) + b^4*(35*A - 3*C) - 48*a^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)","A",9,8,43,0.1860,1,"{3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1054,1,453,0,1.0247149,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a \sin (c+d x) \left(a \left(3 a^2 b B-6 a^3 C+10 a b^2 C-7 b^3 B\right)+4 A b^4\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 b^2 (A-8 C)+8 a^3 b B-16 a^4 C-9 a b^3 B+b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+8 a^4 b B-16 a^5 C+2 a b^4 (3 A-4 C)+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a \sin (c+d x) \left(a \left(3 a^2 b B-6 a^3 C+10 a b^2 C-7 b^3 B\right)+4 A b^4\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 b^2 (A-8 C)+8 a^3 b B-16 a^4 C-9 a b^3 B+b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+8 a^4 b B-16 a^5 C+2 a b^4 (3 A-4 C)+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 2*a^3*b^2*(A - 14*C) + 2*a*b^4*(3*A - 4*C) - 16*a^5*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*b*B - 9*a*b^3*B - 2*a^2*b^2*(A - 8*C) - 16*a^4*C + b^4*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a*(4*A*b^4 + a*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C))*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",8,8,43,0.1860,1,"{3047, 3031, 3023, 2752, 2663, 2661, 2655, 2653}"
1055,1,359,0,0.629763,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 b B-8 a^3 C+a b^2 (A+9 C)-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b^2 (A+15 C)+2 a^3 b B-8 a^4 C-6 a b^3 B+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sin (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 b B-8 a^3 C+a b^2 (A+9 C)-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b^2 (A+15 C)+2 a^3 b B-8 a^4 C-6 a b^3 B+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*b*B - 3*b^3*B - 8*a^3*C + a*b^2*(A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,41,0.1707,1,"{3031, 3021, 2752, 2663, 2661, 2655, 2653}"
1056,1,333,0,0.4859377,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x) \left(a^2 b B+2 a^3 C-2 a b^2 (2 A+3 C)+3 b^3 B\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C-a b B+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b B+2 a^3 C-2 a b^2 (2 A+3 C)+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sin (c+d x) \left(a^2 b B+2 a^3 C-2 a b^2 (2 A+3 C)+3 b^3 B\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C-a b B+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 b B+2 a^3 C-2 a b^2 (2 A+3 C)+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*b*B - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,35,0.2000,1,"{3021, 2754, 2752, 2663, 2661, 2655, 2653}"
1057,1,401,0,1.2294595,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \sin (c+d x) \left(-a^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-C)+3 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-a^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \sin (c+d x) \left(-a^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-C)+3 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-a^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}",1,"(2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",10,9,41,0.2195,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1058,1,461,0,1.6130917,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{b \sin (c+d x) \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+6 a b^3 B-15 A b^4\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) \left(a^2 (-(3 A-2 C))-2 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(a^2 (-(3 A-2 C))-2 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+6 a b^3 B-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(5 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}","-\frac{b \sin (c+d x) \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+6 a b^3 B-15 A b^4\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) \left(a^2 (-(3 A-2 C))-2 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(a^2 (-(3 A-2 C))-2 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+6 a b^3 B-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(5 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"((26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))","A",11,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1059,1,572,0,2.312241,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{b \sin (c+d x) \left(-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left(-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)-20 a b B+35 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^4 d \sqrt{a+b \cos (c+d x)}}-\frac{(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}","-\frac{b \sin (c+d x) \left(-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left(-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)-20 a b B+35 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^4 d \sqrt{a+b \cos (c+d x)}}-\frac{(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}",1,"((105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(12*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((35*A*b^2 - 20*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^4*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*(a + b*Cos[c + d*x])^(3/2)) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^(3/2))","A",12,9,43,0.2093,1,"{3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1060,1,449,0,0.7364595,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \sin (c+d x) \left(-a^2 b^2 (23 A+19 C)+3 a^3 b B+2 a^4 C+29 a b^3 B-3 b^4 (3 A+5 C)\right)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(3 a^2 b B+2 a^3 C-2 a b^2 (4 A+5 C)+5 b^3 B\right)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}+\frac{2 \left(3 a^2 b B+2 a^3 C-2 a b^2 (4 A+5 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-a^2 b^2 (23 A+19 C)+3 a^3 b B+2 a^4 C+29 a b^3 B-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sin (c+d x) \left(-a^2 b^2 (23 A+19 C)+3 a^3 b B+2 a^4 C+29 a b^3 B-3 b^4 (3 A+5 C)\right)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(3 a^2 b B+2 a^3 C-2 a b^2 (4 A+5 C)+5 b^3 B\right)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}+\frac{2 \left(3 a^2 b B+2 a^3 C-2 a b^2 (4 A+5 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-a^2 b^2 (23 A+19 C)+3 a^3 b B+2 a^4 C+29 a b^3 B-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(3*a^3*b*B + 29*a*b^3*B + 2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*a^2*b*B + 5*b^3*B + 2*a^3*C - 2*a*b^2*(4*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) + (2*(3*a^2*b*B + 5*b^3*B + 2*a^3*C - 2*a*b^2*(4*A + 5*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^3*b*B + 29*a*b^3*B + 2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,35,0.2000,1,"{3021, 2754, 2752, 2663, 2661, 2655, 2653}"
1061,1,167,0,0.3421663,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,7,50,0.1400,1,"{3015, 2753, 2752, 2663, 2661, 2655, 2653}"
1062,1,124,0,0.1645088,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,50,0.1200,1,"{24, 2752, 2663, 2661, 2655, 2653}"
1063,1,180,0,0.2890578,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","-\frac{2 b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*(b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,50,0.1400,1,"{24, 2754, 2752, 2663, 2661, 2655, 2653}"
1064,1,271,0,0.4514622,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","-\frac{2 b \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 b \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(4*a*b*B - 5*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(4*a*b*B - 5*a^2*C - 3*b^2*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,50,0.1400,1,"{24, 2754, 2752, 2663, 2661, 2655, 2653}"
1065,1,190,0,0.2554302,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a A+5 a C+5 b B)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a A+5 a C+5 b B)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*(9*A*b + 9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",7,6,41,0.1463,1,"{3033, 3023, 2748, 2635, 2641, 2639}"
1066,1,154,0,0.2384543,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+7 A b+5 b C)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+7 A b+5 b C)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(2*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,6,41,0.1463,1,"{3033, 3023, 2748, 2639, 2635, 2641}"
1067,1,116,0,0.2174688,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,41,0.1220,1,"{3033, 3023, 2748, 2641, 2639}"
1068,1,107,0,0.2242383,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(b*B - a*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,41,0.1220,1,"{3031, 3023, 2748, 2641, 2639}"
1069,1,111,0,0.2376219,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(A*b + a*B - b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*b*B + a*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*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",5,5,41,0.1220,1,"{3031, 3021, 2748, 2641, 2639}"
1070,1,152,0,0.2554272,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sin (c+d x) (3 a A+5 a C+5 b B)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sin (c+d x) (3 a A+5 a C+5 b B)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*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*(3*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,6,41,0.1463,1,"{3031, 3021, 2748, 2636, 2639, 2641}"
1071,1,190,0,0.2818184,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sin (c+d x) (5 a A+7 a C+7 b B)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (3 a B+3 A b+5 b C)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sin (c+d x) (5 a A+7 a C+7 b B)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (3 a B+3 A b+5 b C)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(3*A*b + 3*a*B + 5*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,41,0.1463,1,"{3031, 3021, 2748, 2636, 2641, 2639}"
1072,1,305,0,0.6076571,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(11*A*b^2 + 22*a*b*B + 4*a^2*C + 9*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 4*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)","A",8,7,43,0.1628,1,"{3049, 3033, 3023, 2748, 2635, 2641, 2639}"
1073,1,251,0,0.5370675,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 4*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)","A",7,7,43,0.1628,1,"{3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1074,1,203,0,0.5058477,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 4*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)","A",6,6,43,0.1395,1,"{3049, 3033, 3023, 2748, 2641, 2639}"
1075,1,189,0,0.519033,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (6 a A-2 a C-b B)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (6 a A-2 a C-b B)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a*A - b*B - 2*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,43,0.1395,1,"{3047, 3033, 3023, 2748, 2641, 2639}"
1076,1,180,0,0.5008615,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*(a^2*B - b^2*B + 2*a*b*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(4*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,43,0.1395,1,"{3047, 3031, 3023, 2748, 2641, 2639}"
1077,1,200,0,0.5393731,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (5 a B+4 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (5 a B+4 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(4*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(4*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,6,43,0.1395,1,"{3047, 3031, 3021, 2748, 2641, 2639}"
1078,1,248,0,0.5601134,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (7 a B+4 A b) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (7 a B+4 A b) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,7,43,0.1628,1,"{3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1079,1,302,0,0.6380183,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(4*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*d*Cos[c + d*x]^(5/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,7,43,0.1628,1,"{3047, 3031, 3021, 2748, 2636, 2641, 2639}"
1080,1,361,0,0.9201623,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}",1,"(2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(11*b*B + 6*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)","A",8,7,43,0.1628,1,"{3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1081,1,296,0,0.838873,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}",1,"(2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)","A",7,6,43,0.1395,1,"{3049, 3033, 3023, 2748, 2641, 2639}"
1082,1,279,0,0.83112,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (35 a A-11 a C-7 b B)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (35 a A-11 a C-7 b B)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}",1,"(2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*b*B - 6*a^2*(7*A - 3*C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b^2*(35*a*A - 7*b*B - 11*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,43,0.1628,1,"{3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1083,1,271,0,0.8584572,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (15 a B+35 A b-3 b C)}{15 d}+\frac{2 (a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (15 a B+35 A b-3 b C)}{15 d}+\frac{2 (a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a^2*B - b^2*B + 3*a*b*(5*A - C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(35*A*b + 15*a*B - 3*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,6,43,0.1395,1,"{3047, 3033, 3023, 2748, 2641, 2639}"
1084,1,273,0,0.8236533,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}+\frac{2 a \sin (c+d x) \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+9 A b-5 b C)}{15 d}+\frac{2 (5 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}+\frac{2 a \sin (c+d x) \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+9 A b-5 b C)}{15 d}+\frac{2 (5 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(24*A*b^2 + 35*a*b*B + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,6,43,0.1395,1,"{3047, 3031, 3023, 2748, 2641, 2639}"
1085,1,294,0,0.8510408,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 a \sin (c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(21 a^2 b (3 A+5 C)+21 a^3 B+98 a b^2 B+24 A b^3\right)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 a \sin (c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(21 a^2 b (3 A+5 C)+21 a^3 B+98 a b^2 B+24 A b^3\right)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(24*A*b^3 + 21*a^3*B + 98*a*b^2*B + 21*a^2*b*(3*A + 5*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (2*(6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,6,43,0.1395,1,"{3047, 3031, 3021, 2748, 2641, 2639}"
1086,1,357,0,0.923861,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,7,43,0.1628,1,"{3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1087,1,477,0,1.3160382,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}",1,"(2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (2*(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)","A",9,7,43,0.1628,1,"{3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1088,1,404,0,1.2590206,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}",1,"(2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)","A",8,6,43,0.1395,1,"{3049, 3033, 3023, 2748, 2641, 2639}"
1089,1,379,0,1.2709714,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}",1,"(2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(117*a^2*b*B + 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b^2*(162*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) - (2*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,7,43,0.1628,1,"{3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1090,1,373,0,1.2577353,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(3 a^2 b (49 A-13 C)+42 a^3 B-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac{2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(3 a^2 b (49 A-13 C)+42 a^3 B-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac{2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b*(42*a^3*B - 28*a*b^2*B + 3*a^2*b*(49*A - 13*C) - b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b^2*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b*(21*A*b + 7*a*B - b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,7,43,0.1628,1,"{3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1091,1,386,0,1.294996,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(105*a^2*b*B - 5*b^3*B + 4*a*b^2*(33*A - 5*C) + 6*a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(50*a*b*B + b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(16*A*b^2 + 15*a*b*B + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,6,43,0.1395,1,"{3047, 3033, 3023, 2748, 2641, 2639}"
1092,1,383,0,1.2716892,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d}+\frac{2 a \sin (c+d x) \left(a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d}+\frac{2 a \sin (c+d x) \left(a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(192*A*b^3 + 63*a^3*B + 413*a*b^2*B + a^2*(202*A*b + 350*b*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,6,43,0.1395,1,"{3047, 3031, 3023, 2748, 2641, 2639}"
1093,1,401,0,1.3058583,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+756 a^3 b B+1098 a b^3 B+192 A b^4\right)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+756 a^3 b B+1098 a b^3 B+192 A b^4\right)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(192*A*b^4 + 756*a^3*b*B + 1098*a*b^3*B + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",8,6,43,0.1395,1,"{3047, 3031, 3021, 2748, 2641, 2639}"
1094,1,475,0,1.3955971,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(-2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*(8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",9,7,43,0.1628,1,"{3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1095,1,285,0,1.2858006,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (3 A+C)+21 a^3 b B-21 a^4 C+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 b B-5 a^3 C-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}","-\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (3 A+C)+21 a^3 b B-21 a^4 C+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 b B-5 a^3 C-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"(2*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*b^5*d) - (2*a^3*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^5*(a + b)*d) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,6,43,0.1395,1,"{3049, 3059, 2639, 3002, 2641, 2805}"
1096,1,210,0,0.8788594,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"(2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) + (2*(3*a^2*b*B + b^3*B - 3*a^3*C - a*b^2*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^4*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,6,43,0.1395,1,"{3049, 3059, 2639, 3002, 2641, 2805}"
1097,1,147,0,0.6100552,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(2*(b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*(b^2*(3*A + C) - 3*a*(b*B - a*C))*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,43,0.1395,1,"{3049, 3059, 2639, 3002, 2641, 2805}"
1098,1,97,0,0.3138138,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*(b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",5,5,43,0.1163,1,"{3059, 2639, 3002, 2641, 2805}"
1099,1,118,0,0.5275073,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","-\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*A*EllipticE[(c + d*x)/2, 2])/(a*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*b*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])","A",6,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1100,1,158,0,0.8374329,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 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 b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 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 b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*A*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])","A",7,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1101,1,234,0,1.2361098,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 b \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^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 a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 b \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^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 a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (2*b*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])","A",8,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1102,1,318,0,1.744053,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d}+\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d}+\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^4*d) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^3*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Cos[c + d*x]^(3/2)) - (2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c + d*x]])","A",9,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1103,1,445,0,1.5920966,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+15 a^4 b B-21 a^5 C+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^2 b^2 (5 A-8 C)+25 a^3 b B-35 a^4 C-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)+5 a^3 b B-7 a^4 C-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 b B-7 a^3 C-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+15 a^4 b B-21 a^5 C+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^2 b^2 (5 A-8 C)+25 a^3 b B-35 a^4 C-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)+5 a^3 b B-7 a^4 C-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 b B-7 a^3 C-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right)}",1,"-((25*a^3*b*B - 20*a*b^3*B - 3*a^2*b^2*(5*A - 8*C) - 35*a^4*C + 2*b^4*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - a^3*b^2*(9*A - 20*C) - 21*a^5*C + 4*a*b^4*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 + 5*a^3*b*B - 7*a*b^3*B - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^5*(a + b)^2*d) + ((5*a^2*b*B - 2*b^3*B - a*b^2*(3*A - 4*C) - 7*a^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1104,1,343,0,1.1262486,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-16 C)+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a \left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 C-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-16 C)+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a \left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 C-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}",1,"((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - a^2*b^2*(3*A - 16*C) - 15*a^4*C + 2*b^4*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,43,0.1628,1,"{3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1105,1,251,0,0.7198864,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b B-3 a^3 C+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b B-3 a^3 C+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + a*b^2*(A + 4*C))*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{3047, 3059, 2639, 3002, 2641, 2805}"
1106,1,243,0,0.7080969,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 C+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 C+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-(((A*b^2 - a*(b*B - a*C))*EllipticE[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1107,1,306,0,1.0894206,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}-\frac{\sin (c+d x) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}-\frac{\sin (c+d x) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}",1,"((3*A*b^2 - a*b*B - a^2*(2*A - C))*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*EllipticF[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1108,1,392,0,1.5623485,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 C-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 C-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"-(((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1109,1,654,0,2.6388837,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2}",1,"-((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(20*b^5*(a^2 - b^2)^2*d) + ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C) - 189*a^7*C - 24*a*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^6*(a + b)^3*d) + ((35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1110,1,536,0,1.9013788,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-99 a^3 b^3 B+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-99 a^3 b^3 B+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2}",1,"((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - a^4*b^2*(9*A - 223*C) + a^2*b^4*(15*A - 128*C) - 105*a^6*C - 8*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1111,1,423,0,1.3631092,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 b^2 (A+33 C)-5 a^2 b^3 B+3 a^4 b B-15 a^5 C-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 C-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 C-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 C-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 b^2 (A+33 C)-5 a^2 b^3 B+3 a^4 b B-15 a^5 C-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 C-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 C-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 C-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"-((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C - a*b^4*(7*A + 24*C) + a^3*b^2*(A + 33*C))*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{3047, 3059, 2639, 3002, 2641, 2805}"
1112,1,418,0,1.3330771,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (3 A-5 C)+a^3 b B+3 a^4 C-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (3 A-5 C)+a^3 b B+3 a^4 C-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,43,0.1628,1,"{3047, 3055, 3059, 2639, 3002, 2641, 2805}"
1113,1,413,0,1.2754977,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1114,1,502,0,1.8582073,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\sin (c+d x) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}+\frac{\sin (c+d x) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}",1,"-((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*EllipticF[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1115,1,609,0,2.4935393,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \left(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}-\frac{\sin (c+d x) \left(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*EllipticF[(c + d*x)/2, 2])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",9,6,43,0.1395,1,"{3055, 3059, 2639, 3002, 2641, 2805}"
1116,1,586,0,1.794683,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b B+a^3 (-C)-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sin (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}","\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b B+a^3 (-C)-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sin (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}",1,"-((a - b)*Sqrt[a + b]*(8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) + (Sqrt[a + b]*(24*A*b^2 + (a + 2*b)*(6*b*B - 3*a*C + 8*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) + ((8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)","A",8,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1117,1,483,0,1.1533828,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(8*A*b + a*C + 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1118,1,449,0,1.1353148,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{a+b} \cot (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","\frac{\sqrt{a+b} \cot (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"((a - b)*Sqrt[a + b]*(2*A - C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (Sqrt[a + b]*(2*A*b - a*(2*A - 2*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*b*B + a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,45,0.1556,1,"{3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1119,1,407,0,0.8426155,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) - (2*Sqrt[a + b]*(b*(A - 3*B) - a*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,45,0.1333,1,"{3047, 3053, 2809, 2998, 2816, 2994}"
1120,1,360,0,0.9318926,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(2*A*b + a*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))","A",5,5,45,0.1111,1,"{3047, 3055, 2998, 2816, 2994}"
1121,1,447,0,1.3315575,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","-\frac{2 \sin (c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{2 \sin (c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 2*a*b*(3*A - 7*B) + a^2*(25*A - 63*B + 35*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))","A",6,5,45,0.1111,1,"{3047, 3055, 2998, 2816, 2994}"
1122,1,704,0,2.485257,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}","\frac{\sin (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}",1,"-((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4*a*b^2*(60*A + 28*B + 39*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)","A",9,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1123,1,587,0,1.7868859,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sin (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\sin (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"-((a - b)*Sqrt[a + b]*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d) + (Sqrt[a + b]*(3*a^2*C + 4*b^2*(6*A + 3*B + 4*C) + 2*a*b*(24*A + 15*B + 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((2*b*B + a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1124,1,535,0,1.7930547,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{\sin (c+d x) (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{\sin (c+d x) (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(8*a*A - 4*b*B - 5*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) - (Sqrt[a + b]*(a*(8*A - 8*B - 5*C) - 2*b*(8*A + 2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - ((8*a*A - 4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,8,45,0.1778,1,"{3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1125,1,528,0,1.6668767,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{\sin (c+d x) (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{\sin (c+d x) (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((a - b)*Sqrt[a + b]*(8*A*b + 6*a*B - 3*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A - 3*B + 3*C) - a*b*(8*A - 3*(4*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (Sqrt[a + b]*(2*b*B + 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((8*A*b + 6*a*B - 3*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,7,45,0.1556,1,"{3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1126,1,490,0,1.2911485,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) - (2*Sqrt[a + b]*(3*b^2*(A - 5*B) - 2*a*b*(6*A - 10*B + 15*C) + a^2*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,6,45,0.1333,1,"{3047, 3053, 2809, 2998, 2816, 2994}"
1127,1,450,0,1.4318855,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(25 A-63 B+35 C))+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-2 a^2 b (41 A+70 C)-63 a^3 B-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(25 A-63 B+35 C))+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-2 a^2 b (41 A+70 C)-63 a^3 B-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B + 35*C) + 3*a*b*(19*A - 7*B + 35*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",6,5,45,0.1111,1,"{3047, 3055, 2998, 2816, 2994}"
1128,1,550,0,2.0161511,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","-\frac{2 \sin (c+d x) \left(-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","-\frac{2 \sin (c+d x) \left(-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A - 25*B + 63*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",7,5,45,0.1111,1,"{3047, 3055, 2998, 2816, 2994}"
1129,1,834,0,3.7918611,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{C \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sqrt{\cos (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \cot (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d}","\frac{C \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sqrt{\cos (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \cot (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d}",1,"-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(260*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)","A",10,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1130,1,700,0,2.5337908,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\sin (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(5 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\sin (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(5 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}",1,"-((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(12*A + 16*B + 9*C) + 2*a^2*b*(192*A + 132*B + 59*C) + 4*a*b^2*(108*A + 52*B + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",9,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1131,1,647,0,2.3687492,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sin (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}","\frac{\sin (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) - (Sqrt[a + b]*(a^2*(48*A - 48*B - 33*C) - 4*b^2*(6*A + 3*B + 4*C) - 2*a*b*(72*A + 27*B + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) - (b*(8*a*A - 2*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",9,8,45,0.1778,1,"{3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1132,1,622,0,2.2066248,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{\sin (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{\sin (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((a - b)*Sqrt[a + b]*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*a*d) - (Sqrt[a + b]*(a*b*(56*A - 72*B - 27*C) - 6*b^2*(12*A + 2*B + C) - 8*a^2*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) - (Sqrt[a + b]*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b*(8*A*b + 4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",9,8,45,0.1778,1,"{3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1133,1,643,0,2.3618198,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b (17 A-35 B+45 C)-2 a^3 (9 A-5 B+15 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a B+A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 b (17 A-35 B+45 C)-2 a^3 (9 A-5 B+15 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a B+A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((a - b)*Sqrt[a + b]*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (Sqrt[a + b]*(30*A*b^3 - 2*a^3*(9*A - 5*B + 15*C) + 2*a^2*b*(17*A - 35*B + 45*C) - a*b^2*(46*A - 15*(6*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(5*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(A*b + a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",9,7,45,0.1556,1,"{3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1134,1,580,0,1.7979333,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 b (145 A-119 B+245 C)+a^3 (-(25 A-63 B+35 C))-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(5 a^2 b (29 A+49 C)+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 b (145 A-119 B+245 C)+a^3 (-(25 A-63 B+35 C))-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(5 a^2 b (29 A+49 C)+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(2*(a - b)*Sqrt[a + b]*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) - (2*Sqrt[a + b]*(15*b^3*(A - 7*B) - a^3*(25*A - 63*B + 35*C) + a^2*b*(145*A - 119*B + 245*C) - a*b^2*(135*A - 161*B + 315*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 7*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,6,45,0.1333,1,"{3047, 3053, 2809, 2998, 2816, 2994}"
1135,1,552,0,2.0486542,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \sin (c+d x) \left(a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",7,5,45,0.1111,1,"{3047, 3055, 2998, 2816, 2994}"
1136,1,593,0,1.8608576,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sin (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 b B-5 a^3 C-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","\frac{\sin (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 b B-5 a^3 C-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"-((a - b)*Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d) + (Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 12*b^2*B + 15*a^2*C - 10*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) + ((6*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",8,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1137,1,485,0,1.1225064,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"-((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - (Sqrt[a + b]*(8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",7,7,45,0.1556,1,"{3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1138,1,401,0,0.7924519,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"-(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*(2*A*b + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d) - (Sqrt[a + b]*(2*b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",6,6,45,0.1333,1,"{3061, 3053, 2809, 2998, 2816, 2994}"
1139,1,347,0,0.5352755,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (A-B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (A-B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)","A",5,5,45,0.1111,1,"{3053, 2809, 2998, 2816, 2994}"
1140,1,293,0,0.5709144,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} \cot (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{a+b} \cot (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))","A",4,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1141,1,372,0,0.9333857,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*A*b^2 - 2*a*b*(A + 5*B) + a^2*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))","A",5,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1142,1,466,0,1.4145587,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{a+b} \cot (c+d x) \left(2 a^2 b (22 A+7 (B+5 C))+a^3 (25 A-63 B+35 C)-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (44 A b+70 b C)-63 a^3 B-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}-\frac{2 (6 A b-7 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{a+b} \cot (c+d x) \left(2 a^2 b (22 A+7 (B+5 C))+a^3 (25 A-63 B+35 C)-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (44 A b+70 b C)-63 a^3 B-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}-\frac{2 (6 A b-7 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(a - b)*Sqrt[a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d) + (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2*(3*A + 14*B) + a^3*(25*A - 63*B + 35*C) + 2*a^2*b*(22*A + 7*(B + 5*C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Cos[c + d*x]^(3/2))","A",6,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1143,1,473,0,1.4757395,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a B+4 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (B (a+2 b)+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a B+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}","-\frac{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a B+4 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (B (a+2 b)+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a B+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((a - b)*Sqrt[a + b]*(4*A*b + a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(4*A*b + (a + 2*b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(4*a*A*b - a^2*B + 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",8,8,54,0.1481,1,"{3029, 3003, 3061, 3053, 2809, 2998, 2816, 2994}"
1144,1,256,0,0.4950065,"\int \frac{a+a \cos (c+d x)+2 b \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x] + 2*b*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{4 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}","\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{4 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (4*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,46,0.08696,1,"{3061, 2998, 2816, 2994}"
1145,1,660,0,2.0470227,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\cot (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}-\frac{\cot (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}",1,"-((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d) - ((8*A*b^2 - a*b*(12*B - 5*C) + 15*a^2*C - 2*b^2*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)","A",8,8,45,0.1778,1,"{3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1146,1,535,0,1.4561092,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\cot (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\cot (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","\frac{\sin (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\cot (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\cot (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"-(((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d)) + ((2*A*b^2 - a*(b*(2*B - C) - 3*a*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",7,7,45,0.1556,1,"{3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1147,1,436,0,0.8996012,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 \cot (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 \cot (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}",1,"(2*(A*b^2 - a*(b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d) + (2*(A*b + b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{3051, 2809, 2993, 2998, 2816, 2994}"
1148,1,322,0,0.6831551,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 \cot (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}-\frac{2 \cot (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}",1,"(-2*(2*A*b^2 - a*b*B - a^2*(A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(2*A*b + a*(A - B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1149,1,424,0,1.1102922,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 \sin (c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left(-a^2 (5 A b-3 b C)+3 a^3 B-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}","-\frac{2 \sin (c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left(-a^2 (5 A b-3 b C)+3 a^3 B-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}",1,"(2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d) + (2*(8*A*b^2 + 6*a*b*(A - B) + a^2*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",5,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1150,1,545,0,1.7380143,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \sin (c+d x) \left(-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(a^2 (-(A-5 C))-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b}}-\frac{2 \cot (c+d x) \left(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b}}","\frac{2 \sin (c+d x) \left(-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(a^2 (-(A-5 C))-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b}}-\frac{2 \cot (c+d x) \left(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b}}",1,"(-2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^5*Sqrt[a + b]*d) - (2*(48*A*b^3 + 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",6,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1151,1,723,0,2.6300562,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\sin (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(a^2 b^2 (2 B+21 C)+a^3 b (6 B-5 C)-15 a^4 C-a b^3 (2 A+3 (4 B-C))+6 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{\cot (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\sin (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(a^2 b^2 (2 B+21 C)+a^3 b (6 B-5 C)-15 a^4 C-a b^3 (2 A+3 (4 B-C))+6 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{\cot (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) - ((6*A*b^4 - a*b^3*(2*A + 3*(4*B - C)) + a^3*b*(6*B - 5*C) - 15*a^4*C + a^2*b^2*(2*B + 21*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(2*b*B - 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])","A",8,7,45,0.1556,1,"{3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1152,1,589,0,1.6177495,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 C-4 a b^3 B+A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(a^2 b C+3 a^3 C-a b^2 (3 A+B+6 C)+b^3 (A+3 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 C-4 a b^3 B+A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}","\frac{2 \sin (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 C-4 a b^3 B+A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(a^2 b C+3 a^3 C-a b^2 (3 A+B+6 C)+b^3 (A+3 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 C-4 a b^3 B+A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(-2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(b^3*(A + 3*B) + 3*a^3*C + a^2*b*C - a*b^2*(3*A + B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{3047, 3051, 2809, 2993, 2998, 2816, 2994}"
1153,1,457,0,1.1474538,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 \sin (c+d x) \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(a^2 (-(3 A+3 B+C))+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \cot (c+d x) \left(6 a^2 A b+4 a^2 b C-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}","\frac{2 \sin (c+d x) \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(a^2 (-(3 A+3 B+C))+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \cot (c+d x) \left(6 a^2 A b+4 a^2 b C-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*A*b^2 - a^2*(3*A + 3*B + C) + a*b*(3*A + B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,45,0.1111,1,"{3055, 2993, 2998, 2816, 2994}"
1154,1,495,0,1.3195907,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","-\frac{2 \sin (c+d x) \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 C-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(-a^2 b (9 A+3 B+C)-3 a^3 (A-B-C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \cot (c+d x) \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}","-\frac{2 \sin (c+d x) \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 C-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(-a^2 b (9 A+3 B+C)-3 a^3 (A-B-C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \cot (c+d x) \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(8*A*b^3 + 2*a*b^2*(3*A - B) - 3*a^3*(A - B - C) - a^2*b*(9*A + 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1155,1,620,0,2.3014516,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 \sin (c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(10 a^2 A b^2-7 a^3 b B+4 a^4 C+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(-2 a^2 b^2 (8 A+3 B-C)-3 a^3 b (3 A-3 B-C)+a^4 (-(A-3 B+3 C))+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}","\frac{2 \sin (c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(10 a^2 A b^2-7 a^3 b B+4 a^4 C+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(-2 a^2 b^2 (8 A+3 B-C)-3 a^3 b (3 A-3 B-C)+a^4 (-(A-3 B+3 C))+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(-2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(16*A*b^4 + 4*a*b^3*(3*A - 2*B) - 3*a^3*b*(3*A - 3*B - C) - 2*a^2*b^2*(8*A + 3*B - C) - a^4*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))","A",6,4,45,0.08889,1,"{3055, 2998, 2816, 2994}"
1156,1,367,0,0.939783,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+2 a b B \left(m^2+5 m+4\right)+b^2 (m+1) (A (m+4)+C (m+3))\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) \left(a^2 B (m+3)+2 a b (A (m+3)+C (m+2))+b^2 B (m+2)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(2 a^2 C+2 a b B (m+4)+A b^2 (m+4)+b^2 C (m+3)\right)}{d (m+2) (m+4)}+\frac{b \sin (c+d x) (2 a C+b B (m+4)) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+2 a b B \left(m^2+5 m+4\right)+b^2 (m+1) (A (m+4)+C (m+3))\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) \left(a^2 B (m+3)+2 a b (A (m+3)+C (m+2))+b^2 B (m+2)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(2 a^2 C+2 a b B (m+4)+A b^2 (m+4)+b^2 C (m+3)\right)}{d (m+2) (m+4)}+\frac{b \sin (c+d x) (2 a C+b B (m+4)) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}",1,"((2*a^2*C + b^2*C*(3 + m) + A*b^2*(4 + m) + 2*a*b*B*(4 + m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (b*(2*a*C + b*B*(4 + m))*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((2*a*b*B*(4 + 5*m + m^2) + a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - ((b^2*B*(2 + m) + a^2*B*(3 + m) + 2*a*b*(C*(2 + m) + A*(3 + m)))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",6,5,41,0.1220,1,"{3049, 3033, 3023, 2748, 2643}"
1157,1,235,0,0.3735745,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) (a A (m+2)+(m+1) (a C+b B)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) (a B (m+3)+A b (m+3)+b C (m+2)) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{(a C+b B) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) (a A (m+2)+(m+1) (a C+b B)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) (a B (m+3)+A b (m+3)+b C (m+2)) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{(a C+b B) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}",1,"((b*B + a*C)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) + (b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)) - (((b*B + a*C)*(1 + m) + a*A*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - ((b*C*(2 + m) + A*b*(3 + m) + a*B*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",5,4,39,0.1026,1,"{3033, 3023, 2748, 2643}"
1158,1,372,0,0.4208375,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"(a*(A*b^2 - a*(b*B - a*C))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*AppellF1[1/2, -m/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/(b*(a^2 - b^2)*d*(Cos[c + d*x]^2)^(m/2)) - ((b*B - a*C)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (C*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",8,5,41,0.1220,1,"{3063, 2643, 2823, 3189, 429}"
1159,1,564,0,1.0293669,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} \left(a^2 b^2 (A (-m)+A+C (m+2))+a^3 b B m+a^4 (-C) (m+1)-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} \left(a^2 b^2 (A (-m)+A+C (m+2))+a^3 b B m+a^4 (-C) (m+1)-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (-C) (m+1)+a b B m+b^2 (C-A m)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{\sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} \left(a^2 b^2 (A (-m)+A+C (m+2))+a^3 b B m+a^4 (-C) (m+1)-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} \left(a^2 b^2 (A (-m)+A+C (m+2))+a^3 b B m+a^4 (-C) (m+1)-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (-C) (m+1)+a b B m+b^2 (C-A m)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((A*b^4*m + a^3*b*B*m - a*b^3*B*(1 + m) - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)^2*d) - ((A*b^4*m + a^3*b*B*m - a*b^3*B*(1 + m) - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, -m/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/(a*b*(a^2 - b^2)^2*d*(Cos[c + d*x]^2)^(m/2)) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) + ((a*b*B*m - a^2*C*(1 + m) + b^2*(C - A*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) + ((A*b^2 - a*(b*B - a*C))*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*b*(a^2 - b^2)*d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",9,6,41,0.1463,1,"{3055, 3063, 2643, 2823, 3189, 429}"
1160,1,205,0,0.2853309,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,7,33,0.2121,1,"{4221, 3032, 3021, 2748, 2636, 2641, 2639}"
1161,1,172,0,0.2632446,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,33,0.2121,1,"{4221, 3032, 3021, 2748, 2636, 2639, 2641}"
1162,1,135,0,0.2355492,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{4221, 3032, 3021, 2748, 2641, 2639}"
1163,1,135,0,0.2383451,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,33,0.1818,1,"{4221, 3032, 3023, 2748, 2641, 2639}"
1164,1,141,0,0.2327747,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,33,0.1818,1,"{4221, 3034, 3023, 2748, 2641, 2639}"
1165,1,174,0,0.2567204,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",7,7,33,0.2121,1,"{4221, 3034, 3023, 2748, 2639, 2635, 2641}"
1166,1,205,0,0.2796741,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 a (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*a*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4221, 3034, 3023, 2748, 2635, 2641, 2639}"
1167,1,270,0,0.6028938,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (2 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}","\frac{2 a^2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (2 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(-16*a^2*(2*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (16*a^2*(2*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",10,9,35,0.2571,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1168,1,237,0,0.5627328,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}","\frac{2 a^2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}",1,"(-4*a^2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,9,35,0.2571,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1169,1,196,0,0.5419028,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (17 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}-\frac{16 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{2 a^2 (17 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}-\frac{16 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(-16*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,8,35,0.2286,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2641, 2639}"
1170,1,196,0,0.5431326,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{2 a^2 (5 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}","-\frac{2 a^2 (5 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,35,0.2286,1,"{4221, 3044, 2975, 2968, 3023, 2748, 2641, 2639}"
1171,1,200,0,0.5048275,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 a^2 (15 A-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}","-\frac{2 a^2 (15 A-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}",1,"(16*a^2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,35,0.2286,1,"{4221, 3044, 2976, 2968, 3023, 2748, 2641, 2639}"
1172,1,204,0,0.5090971,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}",1,"(4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2641, 2639}"
1173,1,237,0,0.5306043,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (21 A+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 (21 A+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(16*a^2*(3*A + 2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1174,1,270,0,0.5878567,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a^2*(33*A + 25*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1175,1,319,0,0.7686219,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}","\frac{8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(105*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(35*A + 44*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",11,9,35,0.2571,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1176,1,286,0,0.7279188,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{8 a^3 (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}","\frac{8 a^3 (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}",1,"(-4*a^3*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a^3*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(73*A + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",10,9,35,0.2571,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1177,1,253,0,0.6983612,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{8 a^3 (53 A+70 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{12 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}","\frac{8 a^3 (53 A+70 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{12 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(53*A + 70*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (12*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,8,35,0.2286,1,"{4221, 3044, 2975, 2968, 3021, 2748, 2641, 2639}"
1178,1,253,0,0.6820424,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (11 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{4 a^3 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}","-\frac{4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (11 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{4 a^3 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(-4*a^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(11*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,8,35,0.2286,1,"{4221, 3044, 2975, 2968, 3023, 2748, 2641, 2639}"
1179,1,251,0,0.6893317,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}","-\frac{8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(-4*a^3*(5*A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (8*a^3*(10*A - 3*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 3*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,35,0.2571,1,"{4221, 3044, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1180,1,257,0,0.6796206,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}","-\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}",1,"(4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*(7*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,8,35,0.2286,1,"{4221, 3044, 2976, 2968, 3023, 2748, 2641, 2639}"
1181,1,253,0,0.6696098,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}","\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}",1,"(4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2641, 2639}"
1182,1,286,0,0.7073466,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a^3*(44*A + 35*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Sec[c + d*x]^(3/2)) + (2*(33*A + 35*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1183,1,319,0,0.7588095,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{12 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{12 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^3*(221*A + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (40*a^3*(143*A + 118*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (12*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(5/2)) + (2*(143*A + 145*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",11,9,35,0.2571,1,"{4221, 3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1184,1,232,0,0.3263555,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]),x]","\frac{(7 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","\frac{(7 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",8,6,35,0.1714,1,"{4221, 3042, 2748, 2636, 2639, 2641}"
1185,1,190,0,0.2949692,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]),x]","\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{4221, 3042, 2748, 2636, 2641, 2639}"
1186,1,153,0,0.2786631,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{4221, 3042, 2748, 2636, 2639, 2641}"
1187,1,123,0,0.2441599,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","-\frac{(A+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(A+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4221, 3042, 2748, 2641, 2639}"
1188,1,162,0,0.2702682,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{4221, 3042, 2748, 2639, 2635, 2641}"
1189,1,199,0,0.2906659,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4221, 3042, 2748, 2635, 2641, 2639}"
1190,1,232,0,0.3141849,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])","A",8,6,35,0.1714,1,"{4221, 3042, 2748, 2635, 2639, 2641}"
1191,1,229,0,0.4533436,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 (5 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*(5*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((7*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,7,35,0.2000,1,"{4221, 3042, 2978, 2748, 2636, 2641, 2639}"
1192,1,195,0,0.4141396,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(5 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{4 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","-\frac{(5 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{4 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"(-4*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,35,0.2000,1,"{4221, 3042, 2978, 2748, 2636, 2639, 2641}"
1193,1,165,0,0.3941075,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,x]","-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{4221, 3042, 2978, 2748, 2641, 2639}"
1194,1,166,0,0.3889707,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(4*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{4221, 3042, 2977, 2748, 2641, 2639}"
1195,1,201,0,0.4324725,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{2 (A+5 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{2 (A+5 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"-(((A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A + 7*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (2*(A + 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3042, 2977, 2748, 2639, 2635, 2641}"
1196,1,236,0,0.4613639,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(4*(5*A + 14*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A + 3*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (4*(5*A + 14*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",8,7,35,0.2000,1,"{4221, 3042, 2977, 2748, 2635, 2641, 2639}"
1197,1,282,0,0.6188001,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3,x]","\frac{(11 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","\frac{(11 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"((119*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((119*A + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((11*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",9,7,35,0.2000,1,"{4221, 3042, 2978, 2748, 2636, 2641, 2639}"
1198,1,259,0,0.6160236,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3,x]","\frac{(49 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (4 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(49 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (4 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"-((49*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{4221, 3042, 2978, 2748, 2636, 2639, 2641}"
1199,1,224,0,0.5721256,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3,x]","-\frac{(9 A-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (3 A-2 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{(9 A-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (3 A-2 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}",1,"((9*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - (2*(3*A - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A - C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4221, 3042, 2978, 2748, 2641, 2639}"
1200,1,220,0,0.564981,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","-\frac{(A-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(A-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"((A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + (2*(2*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - 9*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3042, 2977, 2978, 2748, 2641, 2639}"
1201,1,218,0,0.5750588,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"-((A - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 13*C)*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4221, 3042, 2977, 2748, 2641, 2639}"
1202,1,249,0,0.5972131,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"-((9*A + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) - (2*C*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A + 119*C)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((A + 11*C)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])","A",8,7,35,0.2000,1,"{4221, 3042, 2977, 2748, 2639, 2635, 2641}"
1203,1,290,0,0.6569422,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(7*(7*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - (2*(A + 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A + 63*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])","A",9,7,35,0.2000,1,"{4221, 3042, 2977, 2748, 2635, 2641, 2639}"
1204,1,213,0,0.58155,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(16*a*(16*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,37,0.1351,1,"{4221, 3044, 2980, 2772, 2771}"
1205,1,168,0,0.5055978,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (24 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (24 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*(24*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,5,37,0.1351,1,"{4221, 3044, 2980, 2772, 2771}"
1206,1,123,0,0.4368674,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (8 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (8 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*(8*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",4,4,37,0.1081,1,"{4221, 3044, 2980, 2771}"
1207,1,136,0,0.4159692,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{4221, 3044, 2980, 2774, 216}"
1208,1,137,0,0.4313101,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,37,0.1351,1,"{4221, 3044, 2981, 2774, 216}"
1209,1,144,0,0.4277126,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (8 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (8 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(8*A + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4221, 3046, 2981, 2774, 216}"
1210,1,189,0,0.5053321,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3046, 2981, 2770, 2774, 216}"
1211,1,234,0,0.5936957,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a (48 A+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{a (48 A+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(48*A + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*C*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,37,0.1622,1,"{4221, 3046, 2981, 2770, 2774, 216}"
1212,1,266,0,0.8361812,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}","\frac{2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}",1,"(16*a^2*(112*A + 143*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(28*A + 33*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2772, 2771}"
1213,1,219,0,0.7480885,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (52 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}","\frac{2 a^2 (52 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}",1,"(4*a^2*(136*A + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2772, 2771}"
1214,1,172,0,0.654635,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{6 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}","\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{6 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}",1,"(2*a^2*(104*A + 175*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,5,37,0.1351,1,"{4221, 3044, 2975, 2980, 2771}"
1215,1,183,0,0.6191451,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}","\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2774, 216}"
1216,1,181,0,0.6397841,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{a^2 (8 A-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}","-\frac{a^2 (8 A-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}",1,"(3*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,37,0.1622,1,"{4221, 3044, 2975, 2981, 2774, 216}"
1217,1,195,0,0.6525375,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^{3/2} (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}","\frac{a^{3/2} (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^(3/2)*(8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,37,0.1622,1,"{4221, 3044, 2976, 2981, 2774, 216}"
1218,1,191,0,0.643296,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^{3/2} (24 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (24 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^(3/2)*(24*A + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3046, 2976, 2981, 2774, 216}"
1219,1,238,0,0.7447974,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 (16 A+13 C) \sin (c+d x)}{32 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^2 (16 A+13 C) \sin (c+d x)}{32 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(3/2)*(112*A + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(16*A + 13*C)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3046, 2976, 2981, 2770, 2774, 216}"
1220,1,285,0,0.8238415,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(3/2)*(176*A + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (3*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,7,37,0.1892,1,"{4221, 3046, 2976, 2981, 2770, 2774, 216}"
1221,1,313,0,1.0560209,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{15}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2),x]","\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}","\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}",1,"(16*a^3*(8368*A + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)","A",8,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2772, 2771}"
1222,1,266,0,0.9578459,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 a^3 (232 A+297 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}","\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 a^3 (232 A+297 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}",1,"(4*a^3*(568*A + 759*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(232*A + 297*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2772, 2771}"
1223,1,219,0,0.8624217,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a^3 (8 A+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}","\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a^3 (8 A+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}",1,"(2*a^3*(584*A + 903*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8*A + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,37,0.1351,1,"{4221, 3044, 2975, 2980, 2771}"
1224,1,230,0,0.7999041,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}","\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}",1,"(2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,37,0.1622,1,"{4221, 3044, 2975, 2980, 2774, 216}"
1225,1,230,0,0.8553868,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{a^3 (64 A+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{5 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}","-\frac{a^3 (64 A+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{5 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(5*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,37,0.1622,1,"{4221, 3044, 2975, 2981, 2774, 216}"
1226,1,238,0,0.8521881,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{10 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{10 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,37,0.1892,1,"{4221, 3044, 2975, 2976, 2981, 2774, 216}"
1227,1,242,0,0.8598433,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}","\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}",1,"(5*a^(5/2)*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,37,0.1622,1,"{4221, 3044, 2976, 2981, 2774, 216}"
1228,1,238,0,0.8519499,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^{5/2} (304 A+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (304 A+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"(a^(5/2)*(304*A + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (5*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])","A",7,6,37,0.1622,1,"{4221, 3046, 2976, 2981, 2774, 216}"
1229,1,285,0,0.9488289,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(5/2)*(400*A + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,7,37,0.1892,1,"{4221, 3046, 2976, 2981, 2770, 2774, 216}"
1230,1,332,0,1.0458741,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(24*A + 23*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sec[c + d*x]^(5/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,7,37,0.1892,1,"{4221, 3046, 2976, 2981, 2770, 2774, 216}"
1231,1,289,0,1.0230442,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,37,0.1622,1,"{4221, 3044, 2984, 12, 2782, 205}"
1232,1,244,0,0.8294546,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (31 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (31 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,37,0.1622,1,"{4221, 3044, 2984, 12, 2782, 205}"
1233,1,201,0,0.6405554,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (13 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (13 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,37,0.1622,1,"{4221, 3044, 2984, 12, 2782, 205}"
1234,1,156,0,0.4627096,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3044, 2984, 12, 2782, 205}"
1235,1,175,0,0.5043762,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3044, 2982, 2782, 205, 2774, 216}"
1236,1,173,0,0.505597,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"-((C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3046, 2982, 2782, 205, 2774, 216}"
1237,1,223,0,0.6872492,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - (C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4221, 3046, 2983, 2982, 2782, 205, 2774, 216}"
1238,1,266,0,0.8799791,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","-\frac{(8 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","-\frac{(8 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((8*A + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) - (C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,8,37,0.2162,1,"{4221, 3046, 2983, 2982, 2782, 205, 2774, 216}"
1239,1,315,0,1.0845474,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}","\frac{(19 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}",1,"((19*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,37,0.1622,1,"{4221, 3042, 2984, 12, 2782, 205}"
1240,1,268,0,0.8819769,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(15 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(49 A+25 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(49 A+25 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \cos (c+d x)+a}}",1,"-((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((49*A + 25*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,37,0.1622,1,"{4221, 3042, 2984, 12, 2782, 205}"
1241,1,221,0,0.7190539,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(11 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}","\frac{(11 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,37,0.1622,1,"{4221, 3042, 2984, 12, 2782, 205}"
1242,1,172,0,0.5347716,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(7 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3042, 2984, 12, 2782, 205}"
1243,1,185,0,0.5567647,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3042, 2982, 2782, 205, 2774, 216}"
1244,1,228,0,0.7221915,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(-3*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4221, 3042, 2983, 2982, 2782, 205, 2774, 216}"
1245,1,285,0,0.9309199,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,8,37,0.2162,1,"{4221, 3042, 2983, 2982, 2782, 205, 2774, 216}"
1246,1,315,0,1.0860297,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(157 A+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(157 A+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-((283*A + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,37,0.1892,1,"{4221, 3042, 2978, 2984, 12, 2782, 205}"
1247,1,266,0,0.9173489,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{5 (19 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{5 (19 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,37,0.1892,1,"{4221, 3042, 2978, 2984, 12, 2782, 205}"
1248,1,219,0,0.7101742,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(49 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{5 (15 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{5 (15 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3042, 2978, 2984, 12, 2782, 205}"
1249,1,174,0,0.5230079,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(19 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3042, 2978, 12, 2782, 205}"
1250,1,232,0,0.7192572,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(5 A-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(5 A-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4221, 3042, 2977, 2982, 2782, 205, 2774, 216}"
1251,1,277,0,0.9345473,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{(3 A+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(3 A+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(A-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(3 A+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(3 A+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(A-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(-5*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,9,37,0.2432,1,"{4221, 3042, 2977, 2983, 2982, 2782, 205, 2774, 216}"
1252,1,334,0,1.1542282,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{(7 A+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(8 A+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(11 A+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(3 A+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(7 A+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(8 A+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(11 A+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(3 A+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((8*A + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",10,9,37,0.2432,1,"{4221, 3042, 2977, 2983, 2982, 2782, 205, 2774, 216}"
1253,1,151,0,0.1376158,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 B \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 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 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 B \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 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 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(-6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,6,30,0.2000,1,"{4221, 3010, 2748, 2636, 2639, 2641}"
1254,1,123,0,0.1222026,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,30,0.2000,1,"{4221, 3010, 2748, 2636, 2641, 2639}"
1255,1,97,0,0.1050662,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,30,0.2000,1,"{4221, 3010, 2748, 2636, 2639, 2641}"
1256,1,75,0,0.0932098,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d","A",5,5,30,0.1667,1,"{4221, 3010, 2748, 2641, 2639}"
1257,1,101,0,0.1057581,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\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)}{d}+\frac{2 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 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)}{d}+\frac{2 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,30,0.2000,1,"{4221, 3010, 2748, 2639, 2635, 2641}"
1258,1,127,0,0.1194193,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]],x]","\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(6*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,30,0.2000,1,"{4221, 3010, 2748, 2635, 2641, 2639}"
1259,1,151,0,0.1302349,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2),x]","\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 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 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 C \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}","\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 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 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 C \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*C*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,6,30,0.2000,1,"{4221, 3010, 2748, 2635, 2639, 2641}"
1260,1,163,0,0.1531705,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}","\frac{2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}",1,"(-2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,31,0.1935,1,"{4221, 3021, 2748, 2636, 2641, 2639}"
1261,1,127,0,0.1346615,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}",1,"(-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{4221, 3021, 2748, 2636, 2639, 2641}"
1262,1,101,0,0.1207651,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}","-\frac{2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}",1,"(-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,31,0.1613,1,"{4221, 3021, 2748, 2641, 2639}"
1263,1,105,0,0.1196212,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{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 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{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 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",5,5,31,0.1613,1,"{4221, 3023, 2748, 2641, 2639}"
1264,1,133,0,0.135555,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]],x]","\frac{2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,31,0.1935,1,"{4221, 3023, 2748, 2639, 2635, 2641}"
1265,1,163,0,0.1567821,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2),x]","\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 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 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 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 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",7,6,31,0.1935,1,"{4221, 3023, 2748, 2635, 2641, 2639}"
1266,1,217,0,0.3518257,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 (B+C)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}","\frac{2 a (5 A+7 (B+C)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(-2*a*(3*A + 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 3*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*(B + C))*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*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,7,41,0.1707,1,"{4221, 3031, 3021, 2748, 2636, 2641, 2639}"
1267,1,179,0,0.3158616,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (3 A+5 (B+C)) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 a (3 A+5 (B+C)) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*a*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*(B + C))*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*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,41,0.1707,1,"{4221, 3031, 3021, 2748, 2636, 2639, 2641}"
1268,1,140,0,0.3126374,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 a (A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 a (A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*a*(A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4221, 3031, 3021, 2748, 2641, 2639}"
1269,1,141,0,0.2621052,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*a*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,41,0.1463,1,"{4221, 3031, 3023, 2748, 2641, 2639}"
1270,1,147,0,0.2699487,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 a (3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a (3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*a*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(B + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,41,0.1463,1,"{4221, 3033, 3023, 2748, 2641, 2639}"
1271,1,184,0,0.2979672,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a (7 A+7 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a (7 A+7 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a*(5*A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(B + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 7*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",7,7,41,0.1707,1,"{4221, 3033, 3023, 2748, 2639, 2635, 2641}"
1272,1,217,0,0.3450414,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*a*(9*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(B + C)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 9*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*(B + C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4221, 3033, 3023, 2748, 2635, 2641, 2639}"
1273,1,291,0,0.6518966,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}","\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(-4*a^2*(8*A + 9*B + 12*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(8*A + 9*B + 12*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*(4*A + 9*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",10,9,43,0.2093,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1274,1,255,0,0.630896,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (6 A+7 B+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}","\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (6 A+7 B+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}",1,"(-4*a^2*(3*A + 4*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 4*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 49*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,9,43,0.2093,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1275,1,214,0,0.6085193,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 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) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 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) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(-4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(4*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,8,43,0.1860,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2641, 2639}"
1276,1,212,0,0.5886003,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (2 A+3 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}","-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (2 A+3 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(4*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,43,0.1860,1,"{4221, 3043, 2975, 2968, 3023, 2748, 2641, 2639}"
1277,1,212,0,0.58893,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 B+4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}","-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 B+4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}",1,"(4*a^2*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,43,0.1860,1,"{4221, 3043, 2976, 2968, 3023, 2748, 2641, 2639}"
1278,1,219,0,0.5789588,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (14 A+7 B+6 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (14 A+7 B+6 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}",1,"(4*a^2*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*(7*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]])","A",8,8,43,0.1860,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2641, 2639}"
1279,1,255,0,0.6113048,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(4*a^2*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (2*(9*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,9,43,0.2093,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1280,1,291,0,0.6503697,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (66 A+55 B+50 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (66 A+55 B+50 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^2*(9*A + 8*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 121*B + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(11*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 8*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(66*A + 55*B + 50*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,43,0.2093,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1281,1,343,0,0.8432384,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}","\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(-4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*A + 17*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*(6*A + 11*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",11,9,43,0.2093,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1282,1,307,0,0.8157765,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+13 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (2 A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}","\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+13 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (2 A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}",1,"(-4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 21*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(32*A + 41*B + 42*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",10,9,43,0.2093,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1283,1,271,0,0.7755158,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}","\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(-4*a^3*(7*A + 9*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(6*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,8,43,0.1860,1,"{4221, 3043, 2975, 2968, 3021, 2748, 2641, 2639}"
1284,1,270,0,0.7909646,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}","-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(-4*a^3*(9*A + 5*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(33*A + 35*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,8,43,0.1860,1,"{4221, 3043, 2975, 2968, 3023, 2748, 2641, 2639}"
1285,1,267,0,0.7622452,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}","-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(-4*a^3*(5*A - 5*B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(35*A + 15*B - 3*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,43,0.2093,1,"{4221, 3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1286,1,269,0,0.7618664,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+21 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}","-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+21 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}",1,"(4*a^3*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*(7*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 7*B - 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,8,43,0.1860,1,"{4221, 3043, 2976, 2968, 3023, 2748, 2641, 2639}"
1287,1,271,0,0.7842668,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}","\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}",1,"(4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*(3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 99*B + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2641, 2639}"
1288,1,307,0,0.8268134,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*(11*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d*Sec[c + d*x]^(3/2)) + (2*(99*A + 143*B + 105*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 121*B + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,43,0.2093,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1289,1,343,0,0.8451125,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 (13 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 (13 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^3*(221*A + 195*B + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(286*A + 273*B + 236*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*(13*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(5/2)) + (2*(143*A + 195*B + 145*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 195*B + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 105*B + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",11,9,43,0.2093,1,"{4221, 3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1290,1,250,0,0.3766784,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]),x]","\frac{(7 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A-5 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A-5 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","\frac{(7 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A-5 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A-5 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-3*(7*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A - 5*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A - 5*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{4221, 3041, 2748, 2636, 2639, 2641}"
1291,1,205,0,0.3480673,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]),x]","\frac{(5 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A-3 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(5 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A-3 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((3*A - 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A - 3*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{4221, 3041, 2748, 2636, 2641, 2639}"
1292,1,165,0,0.3241946,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]),x]","\frac{(3 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(3 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{4221, 3041, 2748, 2636, 2639, 2641}"
1293,1,130,0,0.2943137,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",5,5,43,0.1163,1,"{4221, 3041, 2748, 2641, 2639}"
1294,1,174,0,0.3184282,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{(3 A-3 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(3 A-3 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 3*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",6,6,43,0.1395,1,"{4221, 3041, 2748, 2639, 2635, 2641}"
1295,1,214,0,0.3395585,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{(5 A-5 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-5 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","\frac{(5 A-5 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-5 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(3*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A - 5*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",7,6,43,0.1395,1,"{4221, 3041, 2748, 2635, 2641, 2639}"
1296,1,250,0,0.3654524,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-3*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A - 7*B + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A - 7*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A - 7*B + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])","A",8,6,43,0.1395,1,"{4221, 3041, 2748, 2635, 2639, 2641}"
1297,1,251,0,0.5144237,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2,x]","\frac{(10 A-5 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(10 A-5 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A - 4*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A - 4*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((10*A - 5*B + 2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,7,43,0.1628,1,"{4221, 3041, 2978, 2748, 2636, 2641, 2639}"
1298,1,215,0,0.4815776,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(5 A-2 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","-\frac{(5 A-2 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"-(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - 2*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,43,0.1628,1,"{4221, 3041, 2978, 2748, 2636, 2639, 2641}"
1299,1,173,0,0.4364091,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,x]","\frac{(2 A+B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}","\frac{(2 A+B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])","A",6,6,43,0.1395,1,"{4221, 3041, 2978, 2748, 2641, 2639}"
1300,1,179,0,0.4444474,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A+2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(B-4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A+2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(B-4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"-(((B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",6,6,43,0.1395,1,"{4221, 3041, 2977, 2748, 2641, 2639}"
1301,1,220,0,0.4967932,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{(2 A-5 B+10 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-4 B+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{(2 A-5 B+10 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(2 A-5 B+10 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-4 B+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{(2 A-5 B+10 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"-(((A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A - 4*B + 7*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((2*A - 5*B + 10*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3041, 2977, 2748, 2639, 2635, 2641}"
1302,1,254,0,0.5205303,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((20*A - 35*B + 56*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A - 2*B + 3*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A - 2*B + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3041, 2977, 2748, 2635, 2641, 2639}"
1303,1,310,0,0.7105944,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3,x]","\frac{(33 A-13 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","\frac{(33 A-13 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"((119*A - 49*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((119*A - 49*B + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4221, 3041, 2978, 2748, 2636, 2641, 2639}"
1304,1,277,0,0.6880303,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3,x]","\frac{(49 A-9 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(49 A-9 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"-((49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4221, 3041, 2978, 2748, 2636, 2639, 2641}"
1305,1,233,0,0.6589358,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3,x]","-\frac{(9 A+B-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(6 A-B-4 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{(9 A+B-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(6 A-B-4 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}",1,"((9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - ((6*A - B - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A + B - C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,6,43,0.1395,1,"{4221, 3041, 2978, 2748, 2641, 2639}"
1306,1,231,0,0.6732767,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","-\frac{(A-B-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(A-B-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"((A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + ((4*A + B - 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - B - 9*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3041, 2977, 2978, 2748, 2641, 2639}"
1307,1,235,0,0.6504194,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{(A+3 B-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 B-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{(A+3 B-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 B-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"-((A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 3*B - 13*C)*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,6,43,0.1395,1,"{4221, 3041, 2977, 2748, 2641, 2639}"
1308,1,272,0,0.6851655,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"-((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) + ((B - 2*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 13*B + 33*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3041, 2977, 2748, 2639, 2635, 2641}"
1309,1,313,0,0.7214881,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{7 (7 A-17 B+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(2 A-7 B+12 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{7 (7 A-17 B+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(2 A-7 B+12 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(7*(7*A - 17*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - ((2*A - 7*B + 12*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A - 17*B + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])","A",9,7,43,0.1628,1,"{4221, 3041, 2977, 2748, 2635, 2641, 2639}"
1310,1,226,0,0.6461447,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}","\frac{2 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(16*a*(16*A + 18*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,45,0.1111,1,"{4221, 3043, 2980, 2772, 2771}"
1311,1,178,0,0.5923774,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}","\frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(4*a*(24*A + 28*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,5,45,0.1111,1,"{4221, 3043, 2980, 2772, 2771}"
1312,1,130,0,0.5003116,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (8 A+10 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}","\frac{2 a (8 A+10 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(2*a*(8*A + 10*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",4,4,45,0.08889,1,"{4221, 3043, 2980, 2771}"
1313,1,140,0,0.4733197,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{4221, 3043, 2980, 2774, 216}"
1314,1,141,0,0.4861667,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} (2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} (2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*(2*B + C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,45,0.1111,1,"{4221, 3043, 2981, 2774, 216}"
1315,1,151,0,0.4863979,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (8 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a} (8 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a]*(8*A + 4*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*(4*B + C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4221, 3045, 2981, 2774, 216}"
1316,1,199,0,0.5626608,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (8 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\sqrt{a} (8 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a]*(8*A + 6*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*(6*B + C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 6*B + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3045, 2981, 2770, 2774, 216}"
1317,1,247,0,0.6534423,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a (48 A+40 B+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a (48 A+40 B+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[a]*(48*A + 40*B + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*(8*B + C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,45,0.1333,1,"{4221, 3045, 2981, 2770, 2774, 216}"
1318,1,284,0,0.9074898,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}","\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(16*a^2*(336*A + 374*B + 429*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(84*A + 110*B + 99*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2772, 2771}"
1319,1,232,0,0.809637,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}","\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}",1,"(4*a^2*(136*A + 156*B + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 72*B + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2772, 2771}"
1320,1,184,0,0.7142901,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}",1,"(2*a^2*(104*A + 126*B + 175*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,5,45,0.1111,1,"{4221, 3043, 2975, 2980, 2771}"
1321,1,192,0,0.6613379,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2774, 216}"
1322,1,191,0,0.7176272,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}","-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(3/2)*(2*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A + 6*B - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,45,0.1333,1,"{4221, 3043, 2975, 2981, 2774, 216}"
1323,1,201,0,0.7173703,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^{3/2} (8 A+12 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}","\frac{a^{3/2} (8 A+12 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^(3/2)*(8*A + 12*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,45,0.1333,1,"{4221, 3043, 2976, 2981, 2774, 216}"
1324,1,201,0,0.7269319,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^{3/2} (24 A+14 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (24 A+14 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^(3/2)*(24*A + 14*B + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*(2*B + C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3045, 2976, 2981, 2774, 216}"
1325,1,253,0,0.8066432,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 (48 A+56 B+39 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^2 (48 A+56 B+39 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(3/2)*(112*A + 88*B + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(48*A + 56*B + 39*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(8*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 88*B + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3045, 2976, 2981, 2770, 2774, 216}"
1326,1,303,0,0.9075699,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(3/2)*(176*A + 150*B + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(10*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,7,45,0.1556,1,"{4221, 3045, 2976, 2981, 2770, 2774, 216}"
1327,1,334,0,1.1761716,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{15}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2),x]","\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}","\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(16*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2522*B + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)","A",8,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2772, 2771}"
1328,1,284,0,1.0584097,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}","\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}",1,"(4*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(1160*A + 1364*B + 1485*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2772, 2771}"
1329,1,234,0,0.9533841,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}","\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}",1,"(2*a^3*(584*A + 690*B + 903*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8*A + 10*B + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 90*B + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*a*(5*A + 9*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,45,0.1111,1,"{4221, 3043, 2975, 2980, 2771}"
1330,1,242,0,0.8781969,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (5 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,45,0.1333,1,"{4221, 3043, 2975, 2980, 2774, 216}"
1331,1,243,0,0.9524026,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{a^{5/2} (2 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}","-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{a^{5/2} (2 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^(5/2)*(2*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 70*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,45,0.1333,1,"{4221, 3043, 2975, 2981, 2774, 216}"
1332,1,253,0,0.948697,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^{5/2} (8 A+20 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}","\frac{a^{5/2} (8 A+20 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}",1,"(a^(5/2)*(8*A + 20*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,45,0.1556,1,"{4221, 3043, 2975, 2976, 2981, 2774, 216}"
1333,1,251,0,0.9540692,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}","\frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^(5/2)*(40*A + 38*B + 25*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,45,0.1333,1,"{4221, 3043, 2976, 2981, 2774, 216}"
1334,1,253,0,0.9407519,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{a (8 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{a (8 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"(a^(5/2)*(304*A + 200*B + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (a*(8*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])","A",7,6,45,0.1333,1,"{4221, 3045, 2976, 2981, 2774, 216}"
1335,1,301,0,1.0382274,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(5/2)*(400*A + 326*B + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*(2*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,7,45,0.1556,1,"{4221, 3045, 2976, 2981, 2770, 2774, 216}"
1336,1,353,0,1.1588534,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{480 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{480 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (12 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(680*A + 628*B + 545*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(120*A + 156*B + 115*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d*Sec[c + d*x]^(5/2)) + (a*(12*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,7,45,0.1556,1,"{4221, 3045, 2976, 2981, 2770, 2774, 216}"
1337,1,305,0,1.1319705,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (19 A-3 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (19 A-3 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,45,0.1333,1,"{4221, 3043, 2984, 12, 2782, 205}"
1338,1,257,0,0.9097439,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (31 A-7 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (31 A-7 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,45,0.1333,1,"{4221, 3043, 2984, 12, 2782, 205}"
1339,1,211,0,0.7170595,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (13 A-5 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (13 A-5 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,45,0.1333,1,"{4221, 3043, 2984, 12, 2782, 205}"
1340,1,163,0,0.5069077,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3043, 2984, 12, 2782, 205}"
1341,1,178,0,0.5574698,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3043, 2982, 2782, 205, 2774, 216}"
1342,1,181,0,0.557908,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((2*B - C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3045, 2982, 2782, 205, 2774, 216}"
1343,1,235,0,0.7776419,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{(8 A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{(8 A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"((8*A - 4*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*B - C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4221, 3045, 2983, 2982, 2782, 205, 2774, 216}"
1344,1,281,0,1.040547,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","-\frac{(8 A-14 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","-\frac{(8 A-14 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((8*A - 14*B + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + ((6*B - C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A - 2*B + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4221, 3045, 2983, 2982, 2782, 205, 2774, 216}"
1345,1,192,0,0.6990606,"\int \frac{\left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(2 a B+2 A b-b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} (a-b) (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(2 a B+2 A b-b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} (a-b) (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((2*A*b + 2*a*B - b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (b*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,54,0.1296,1,"{4221, 3045, 2982, 2782, 205, 2774, 216}"
1346,1,333,0,1.226497,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A-15 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A-7 B+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A-63 B+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A-273 B+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A-1029 B+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}","\frac{(19 A-15 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A-7 B+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A-63 B+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A-273 B+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A-1029 B+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}",1,"((19*A - 15*B + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,45,0.1333,1,"{4221, 3041, 2984, 12, 2782, 205}"
1347,1,283,0,0.980823,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(15 A-11 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(39 A-35 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 a d \sqrt{a \cos (c+d x)+a}}+\frac{(147 A-95 B+75 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A-11 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(39 A-35 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 a d \sqrt{a \cos (c+d x)+a}}+\frac{(147 A-95 B+75 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}",1,"-((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((147*A - 95*B + 75*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,45,0.1333,1,"{4221, 3041, 2984, 12, 2782, 205}"
1348,1,233,0,0.7730328,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(11 A-7 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}","\frac{(11 A-7 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,45,0.1333,1,"{4221, 3041, 2984, 12, 2782, 205}"
1349,1,181,0,0.5685721,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(7 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3041, 2984, 12, 2782, 205}"
1350,1,189,0,0.5985614,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 A+B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A+B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3041, 2982, 2782, 205, 2774, 216}"
1351,1,242,0,0.8068469,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(A-5 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(A-5 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((2*B - 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A - B + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4221, 3041, 2983, 2982, 2782, 205, 2774, 216}"
1352,1,300,0,1.051108,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{(8 A-12 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A-6 B+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(8 A-12 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A-6 B+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((8*A - 12*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A - B + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A - 6*B + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4221, 3041, 2983, 2982, 2782, 205, 2774, 216}"
1353,1,333,0,1.2413157,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-((283*A - 163*B + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A - 85*B + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,45,0.1556,1,"{4221, 3041, 2978, 2984, 12, 2782, 205}"
1354,1,281,0,1.0180557,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(95 A-39 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-75 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A-9 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(95 A-39 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-75 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(17 A-9 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A - 147*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,45,0.1556,1,"{4221, 3041, 2978, 2984, 12, 2782, 205}"
1355,1,231,0,0.8046076,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(49 A-9 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A-9 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3041, 2978, 2984, 12, 2782, 205}"
1356,1,183,0,0.5730248,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(19 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3041, 2978, 12, 2782, 205}"
1357,1,241,0,0.7790737,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(5 A+3 B-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(5 A+3 B-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A + 3*B - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4221, 3041, 2977, 2982, 2782, 205, 2774, 216}"
1358,1,294,0,1.035662,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{(3 A-11 B+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(3 A-43 B+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(A+7 B-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(3 A-11 B+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(3 A-43 B+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(A+7 B-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((2*B - 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A + 7*B - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4221, 3041, 2977, 2983, 2982, 2782, 205, 2774, 216}"
1359,1,352,0,1.273677,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{(7 A-15 B+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(8 A-20 B+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(11 A-35 B+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 B+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(3 A-11 B+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(7 A-15 B+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(8 A-20 B+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(11 A-35 B+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 B+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(3 A-11 B+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((8*A - 20*B + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A - 115*B + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A - 11*B + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A - 35*B + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",10,9,45,0.2000,1,"{4221, 3041, 2977, 2983, 2982, 2782, 205, 2774, 216}"
1360,1,205,0,0.2956105,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*b*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*A*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,7,33,0.2121,1,"{4221, 3032, 3021, 2748, 2636, 2641, 2639}"
1361,1,172,0,0.2671298,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,33,0.2121,1,"{4221, 3032, 3021, 2748, 2636, 2639, 2641}"
1362,1,135,0,0.2420308,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(-2*b*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{4221, 3032, 3021, 2748, 2641, 2639}"
1363,1,135,0,0.2350751,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,33,0.1818,1,"{4221, 3032, 3023, 2748, 2641, 2639}"
1364,1,141,0,0.2318089,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,33,0.1818,1,"{4221, 3034, 3023, 2748, 2641, 2639}"
1365,1,174,0,0.2561321,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",7,7,33,0.2121,1,"{4221, 3034, 3023, 2748, 2639, 2635, 2641}"
1366,1,205,0,0.2942561,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4221, 3034, 3023, 2748, 2635, 2641, 2639}"
1367,1,292,0,0.6298943,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \left(a^2 (7 A+9 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\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{4 a b (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}","\frac{2 \left(a^2 (7 A+9 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\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{4 a b (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(-2*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a*b*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a*b*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (8*a*A*b*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,8,35,0.2286,1,"{4221, 3048, 3031, 3021, 2748, 2636, 2641, 2639}"
1368,1,243,0,0.568128,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\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{4 a b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}","\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\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{4 a b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(-4*a*b*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a*b*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(4*A*b^2 + a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (8*a*A*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,8,35,0.2286,1,"{4221, 3048, 3031, 3021, 2748, 2636, 2639, 2641}"
1369,1,209,0,0.5280221,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\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{4 a b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\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{4 a b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(-2*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(4*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,35,0.2000,1,"{4221, 3048, 3031, 3021, 2748, 2641, 2639}"
1370,1,194,0,0.5202989,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\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{4 a b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 a A b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\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{4 a b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 a A b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-4*a*b*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,35,0.2000,1,"{4221, 3048, 3031, 3023, 2748, 2641, 2639}"
1371,1,206,0,0.5282842,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\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{4 a b (3 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\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{4 a b (3 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*(5*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(5*A - C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) - (4*a*b*(3*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,7,35,0.2000,1,"{4221, 3048, 3033, 3023, 2748, 2641, 2639}"
1372,1,211,0,0.5132272,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\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{4 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}","\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\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{4 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}",1,"(4*a*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*C*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(4*a^2*C + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3050, 3033, 3023, 2748, 2641, 2639}"
1373,1,245,0,0.5535826,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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{4 a b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a b C \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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{4 a b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a b C \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*C*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(4*a^2*C + b^2*(9*A + 7*C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (4*a*b*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3050, 3033, 3023, 2748, 2639, 2635, 2641}"
1374,1,294,0,0.6334855,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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{4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a b C \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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{4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a b C \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a*b*C*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (4*a*b*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4221, 3050, 3033, 3023, 2748, 2635, 2641, 2639}"
1375,1,333,0,0.9528321,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{63 d}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\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 \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\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 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}+\frac{4 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}","\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{63 d}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\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 \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\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 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}+\frac{4 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}",1,"(-2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,9,35,0.2571,1,"{4221, 3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1376,1,283,0,0.8618427,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d}+\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\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 \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\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 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}+\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}","\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d}+\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\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 \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\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 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}+\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}",1,"(-2*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (6*b*(8*A*b^2 + 7*a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (12*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,8,35,0.2286,1,"{4221, 3048, 3047, 3031, 3021, 2748, 2641, 2639}"
1377,1,269,0,0.8168242,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\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 \left(a^2 (3 A+5 C)+15 b^2 (A-C)\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{4 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}-\frac{2 b^3 (9 A-5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}","\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\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 \left(a^2 (3 A+5 C)+15 b^2 (A-C)\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{4 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}-\frac{2 b^3 (9 A-5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}",1,"(-2*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^3*(9*A - 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(8*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,8,35,0.2286,1,"{4221, 3048, 3047, 3031, 3023, 2748, 2641, 2639}"
1378,1,258,0,0.8274401,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\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 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\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 b^2 (5 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{4 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^3 (35 A-3 C) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\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 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\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 b^2 (5 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{4 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^3 (35 A-3 C) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*b*(15*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^3*(35*A - 3*C)*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*a*b^2*(5*A - C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,35,0.2286,1,"{4221, 3048, 3047, 3033, 3023, 2748, 2641, 2639}"
1379,1,284,0,0.8954516,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\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 \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\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 b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}","-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\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 \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\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 b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}",1,"(-2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*a*b^2*(35*A - 11*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(7*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,35,0.2286,1,"{4221, 3048, 3049, 3033, 3023, 2748, 2641, 2639}"
1380,1,285,0,0.8426253,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\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 \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}","\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\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 \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}",1,"(2*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(24*a^2*C + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*a*(63*A*b^2 + 8*a^2*C + 45*b^2*C)*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (4*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3050, 3049, 3033, 3023, 2748, 2641, 2639}"
1381,1,335,0,0.9273431,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{33 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{33 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C)*Sin[c + d*x])/(165*d*Sec[c + d*x]^(3/2)) + (4*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(33*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4221, 3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1382,1,386,0,1.0163449,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a \left(8 a^2 C+143 A b^2+117 b^2 C\right) \sin (c+d x)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\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 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{12 a C \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a \left(8 a^2 C+143 A b^2+117 b^2 C\right) \sin (c+d x)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\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 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{12 a C \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*a*(11*a^2*(7*A + 5*C) + 15*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(24*a^2*C + 11*b^2*(13*A + 11*C))*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (6*a*(143*A*b^2 + 8*a^2*C + 117*b^2*C)*Sin[c + d*x])/(1001*d*Sec[c + d*x]^(5/2)) + (12*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (2*a*(11*a^2*(7*A + 5*C) + 15*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4221, 3050, 3049, 3033, 3023, 2748, 2635, 2641, 2639}"
1383,1,417,0,1.3754291,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d}+\frac{2 \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\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{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\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 \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}","\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d}+\frac{2 \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\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{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\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 \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}",1,"(-8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 891*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",10,9,35,0.2571,1,"{4221, 3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1384,1,365,0,1.2774498,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+192 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{315 d}+\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 b^4 (A-C)\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 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}","\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+192 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{315 d}+\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 b^4 (A-C)\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 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}",1,"(-2*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(192*A*b^4 + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (4*a*b*(32*A*b^2 + a^2*(101*A + 147*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,8,35,0.2286,1,"{4221, 3048, 3047, 3031, 3021, 2748, 2641, 2639}"
1385,1,356,0,1.3013996,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\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{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\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 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}","\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\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{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\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 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}",1,"(-8*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (4*a*b*(96*A*b^2 + a^2*(101*A + 175*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,8,35,0.2286,1,"{4221, 3048, 3047, 3031, 3023, 2748, 2641, 2639}"
1386,1,361,0,1.2672585,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{5 d}+\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)-b^4 (5 A+3 C)\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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}","-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{5 d}+\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)-b^4 (5 A+3 C)\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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}",1,"(-2*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (4*a*b*(2*b^2*(33*A - 5*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,8,35,0.2286,1,"{4221, 3048, 3047, 3033, 3023, 2748, 2641, 2639}"
1387,1,340,0,1.2608133,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\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{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\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{4 a b^3 (175 A-27 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (21 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{16 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}","-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\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{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\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{4 a b^3 (175 A-27 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (21 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{16 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}",1,"(-8*a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a*b^3*(175*A - 27*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) - (2*b^2*(3*a^2*(49*A - 13*C) - b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(21*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (16*A*b*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,35,0.2571,1,"{4221, 3048, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1388,1,360,0,1.3439145,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\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(-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-b^4 (9 A+7 C)\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 (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}-\frac{2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}","-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\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(-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-b^4 (9 A+7 C)\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 (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}-\frac{2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}",1,"(-2*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(3*a^2*(105*A - 41*C) - 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) - (4*a*b*(a^2*(63*A - 31*C) - 6*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*a*b*(21*A - 5*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,8,35,0.2286,1,"{4221, 3048, 3049, 3033, 3023, 2748, 2641, 2639}"
1389,1,369,0,1.2461339,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\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{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}","\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\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{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\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 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}",1,"(8*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a*b*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(16*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (16*a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4221, 3050, 3049, 3033, 3023, 2748, 2641, 2639}"
1390,1,422,0,1.3240268,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \left(11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\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(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(48*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(3/2)) + (16*a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4221, 3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1391,1,266,0,1.201631,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a^3 d}-\frac{2 \left(a^2 (3 A+5 C)+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 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\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{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a^3 d}-\frac{2 \left(a^2 (3 A+5 C)+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 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\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{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}",1,"(-2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) - (2*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)","A",9,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1392,1,200,0,0.8442944,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 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) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 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) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",1,"(2*A*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",8,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1393,1,172,0,0.6155026,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","-\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",7,7,35,0.2000,1,"{4221, 3056, 3059, 2639, 3002, 2641, 2805}"
1394,1,145,0,0.3691704,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)","A",6,6,35,0.1714,1,"{4221, 3060, 2639, 3002, 2641, 2805}"
1395,1,190,0,0.6327151,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\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^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\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^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"(-2*a*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(3*a^2*C + b^2*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3050, 3059, 2639, 3002, 2641, 2805}"
1396,1,241,0,0.9022245,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","-\frac{2 a \left(C \left(3 a^2+b^2\right)+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 b^4 d}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\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 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}-\frac{2 a C \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 a \left(C \left(3 a^2+b^2\right)+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 b^4 d}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\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 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}-\frac{2 a C \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*a^2*C + b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) - (2*a*(3*A*b^2 + (3*a^2 + b^2)*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^4*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Sec[c + d*x]^(3/2)) - (2*a*C*Sin[c + d*x])/(3*b^2*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3050, 3049, 3059, 2639, 3002, 2641, 2805}"
1397,1,299,0,1.2590763,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 b^3 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\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^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 a C \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 b^3 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\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^4 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 a C \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}",1,"(-2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*d) + (2*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^5*(a + b)*d) + (2*C*Sin[c + d*x])/(7*b*d*Sec[c + d*x]^(5/2)) - (2*a*C*Sin[c + d*x])/(5*b^2*d*Sec[c + d*x]^(3/2)) + (2*(7*a^2*C + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*b^3*d*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4221, 3050, 3049, 3059, 2639, 3002, 2641, 2805}"
1398,1,396,0,1.5627167,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\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^2 d \left(a^2-b^2\right)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\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{\left(-a^2 b^2 (7 A-C)-3 a^4 C+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\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^2 d \left(a^2-b^2\right)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\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{\left(-a^2 b^2 (7 A-C)-3 a^4 C+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"-((b*(5*A*b^2 - a^2*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b*(5*A*b^2 - a^2*(4*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",9,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1399,1,330,0,1.1481673,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^2 C+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 b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}","-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^2 C+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 b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"((3*A*b^2 - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a^2*(2*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1400,1,274,0,0.8054983,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+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 b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+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 b d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 b^2 (A+C)+a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"-(((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3056, 3059, 2639, 3002, 2641, 2805}"
1401,1,277,0,0.7873688,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\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 b^2 (A+5 C)-3 a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\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 b^2 (A+5 C)-3 a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((A*b^2 + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((A*b^4 - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{4221, 3048, 3059, 2639, 3002, 2641, 2805}"
1402,1,352,0,1.1142846,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (3 A-16 C)+15 a^4 C-2 b^4 (3 A+C)\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^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\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(-a^2 b^2 (A-7 C)-5 a^4 C+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}","\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(a^2 b^2 (3 A-16 C)+15 a^4 C-2 b^4 (3 A+C)\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^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\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(-a^2 b^2 (A-7 C)-5 a^4 C+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"-((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) + ((a^2*b^2*(3*A - 16*C) + 15*a^4*C - 2*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3048, 3049, 3059, 2639, 3002, 2641, 2805}"
1403,1,430,0,1.5476847,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left(a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\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^5 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\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^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}","\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left(a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\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^5 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\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^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}",1,"((3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*(a^2 - b^2)*d) - (a*(a^2*b^2*(9*A - 20*C) + 21*a^4*C - 4*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4221, 3048, 3049, 3059, 2639, 3002, 2641, 2805}"
1404,1,554,0,2.2381078,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 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)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 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^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}","\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 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)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 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^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"(b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 - 5*a^4*C - a^2*b^2*(13*A + C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",10,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1405,1,477,0,1.7183248,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 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^2 b d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+3 a^6 C+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}","\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-a^2 b^2 (11 A+3 C)-3 a^4 C+5 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^2 b d \left(a^2-b^2\right)^2}-\frac{\left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+3 a^6 C+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"-((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",9,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1406,1,405,0,1.2545352,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\left(-7 a^2 b^2 (A+C)+a^4 C+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 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 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^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+a^6 (-C)+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}","-\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\left(-7 a^2 b^2 (A+C)+a^4 C+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 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (9 A+5 C)+a^4 (-C)+3 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^2 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+a^6 (-C)+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805}"
1407,1,408,0,1.3180167,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","-\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) \sin (c+d x)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\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^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+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 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)-3 a^6 C+A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}","-\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+A b^4\right) \sin (c+d x)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\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^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 b^2 (5 A+9 C)-3 a^4 C+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 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)-3 a^6 C+A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3048, 3055, 3059, 2639, 3002, 2641, 2805}"
1408,1,405,0,1.340485,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A+11 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{a \left(-a^2 b^2 (A+33 C)+15 a^4 C+b^4 (7 A+24 C)\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^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^2 (A+29 C)-15 a^4 C+b^4 (5 A-8 C)\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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+15 a^6 C+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(a^2 b^2 (3 A+11 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{a \left(-a^2 b^2 (A+33 C)+15 a^4 C+b^4 (7 A+24 C)\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^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^2 (A+29 C)-15 a^4 C+b^4 (5 A-8 C)\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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+15 a^6 C+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"-((b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*C + b^4*(7*A + 24*C) - a^2*b^2*(A + 33*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((3*A*b^4 - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4221, 3048, 3047, 3059, 2639, 3002, 2641, 2805}"
1409,1,493,0,1.8715356,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\left(a^2 b^2 (3 A-61 C)+35 a^4 C-b^4 (21 A-8 C)\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(a^2 b^2 (A+13 C)-7 a^4 C+5 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\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^5 d \left(a^2-b^2\right)^2}-\frac{a \left(a^2 b^2 (3 A-65 C)+35 a^4 C-3 b^4 (3 A-8 C)\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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}","\frac{\left(a^2 b^2 (3 A-61 C)+35 a^4 C-b^4 (21 A-8 C)\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(a^2 b^2 (A+13 C)-7 a^4 C+5 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\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^5 d \left(a^2-b^2\right)^2}-\frac{a \left(a^2 b^2 (3 A-65 C)+35 a^4 C-3 b^4 (3 A-8 C)\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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"-(a*(a^2*b^2*(3*A - 65*C) - 3*b^4*(3*A - 8*C) + 35*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((a^4*b^2*(9*A - 223*C) - a^2*b^4*(15*A - 128*C) + 105*a^6*C + 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + ((5*A*b^4 - 7*a^4*C + a^2*b^2*(A + 13*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((a^2*b^2*(3*A - 61*C) - b^4*(21*A - 8*C) + 35*a^4*C)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",9,9,35,0.2571,1,"{4221, 3048, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1410,1,579,0,2.4271161,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{\left(a^2 b^2 (15 A-101 C)+63 a^4 C-b^4 (45 A-8 C)\right) \sin (c+d x)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left(-5 a^2 b^2 (A-7 C)-21 a^4 C+b^4 (11 A-8 C)\right) \sin (c+d x)}{4 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(-a^2 b^2 (A-15 C)-9 a^4 C+7 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{a \left(-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-63 a^6 C-8 b^6 (3 A+C)\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^6 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-315 a^6 C-8 b^6 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+63 a^6 C+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}","\frac{\left(a^2 b^2 (15 A-101 C)+63 a^4 C-b^4 (45 A-8 C)\right) \sin (c+d x)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left(-5 a^2 b^2 (A-7 C)-21 a^4 C+b^4 (11 A-8 C)\right) \sin (c+d x)}{4 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(-a^2 b^2 (A-15 C)-9 a^4 C+7 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{a \left(-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-63 a^6 C-8 b^6 (3 A+C)\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^6 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-315 a^6 C-8 b^6 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+63 a^6 C+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}",1,"-((a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d) + (a*(a^2*b^4*(33*A - 64*C) - 3*a^4*b^2*(5*A - 43*C) - 63*a^6*C - 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) + ((7*A*b^4 - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((a^2*b^2*(15*A - 101*C) - b^4*(45*A - 8*C) + 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + (a*(b^4*(11*A - 8*C) - 5*a^2*b^2*(A - 7*C) - 21*a^4*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",10,9,35,0.2571,1,"{4221, 3048, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1411,1,544,0,1.952544,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","-\frac{2 \left(6 A b^2-7 a^2 (7 A+9 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 b \left(a^2 (13 A+21 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}","-\frac{2 \left(6 A b^2-7 a^2 (7 A+9 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 b \left(a^2 (13 A+21 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}",1,"(-2*(a - b)*Sqrt[a + b]*(16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^5*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(12*a*A*b^2 + 16*A*b^3 + 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*b*(8*A*b^2 + a^2*(13*A + 21*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a^2*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,37,0.1622,1,"{4221, 3048, 3055, 2998, 2816, 2994}"
1412,1,455,0,1.4200274,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}",1,"(2*(a - b)*b*Sqrt[a + b]*(8*A*b^2 + a^2*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b + 8*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^2 - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,37,0.1622,1,"{4221, 3048, 3055, 2998, 2816, 2994}"
1413,1,385,0,1.049284,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}","-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b + 15*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,37,0.1622,1,"{4221, 3048, 3055, 2998, 2816, 2994}"
1414,1,454,0,0.9371366,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*A*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A*b - a*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,37,0.1892,1,"{4221, 3048, 3053, 2809, 2998, 2816, 2994}"
1415,1,499,0,1.2042476,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}","-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(2*A - C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a*A - 2*A*b - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,37,0.2162,1,"{4221, 3048, 3061, 3053, 2809, 2998, 2816, 2994}"
1416,1,515,0,1.2447634,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*A*b + (a + 2*b)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a^2*C - 4*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)","A",8,8,37,0.2162,1,"{4221, 3050, 3061, 3053, 2809, 2998, 2816, 2994}"
1417,1,613,0,1.7254808,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2*C - 2*a*b*C - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) - (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) - ((3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)","A",9,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1418,1,698,0,2.1740769,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-10 a^2 b C+15 a^3 C+4 a b^2 (12 A+7 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(8 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}-\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-10 a^2 b C+15 a^3 C+4 a b^2 (12 A+7 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(8 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}-\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"-((a - b)*Sqrt[a + b]*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 10*a^2*b*C + 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) - (5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (a*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)","A",10,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1419,1,542,0,1.9509585,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (39 A b+63 b C)-21 a^3 (7 A+9 C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}","\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (39 A b+63 b C)-21 a^3 (7 A+9 C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^2 + 8*A*b^3 - 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (4*b*(2*A*b^2 - a^2*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,7,37,0.1892,1,"{4221, 3048, 3047, 3055, 2998, 2816, 2994}"
1420,1,458,0,1.4231834,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{6 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{6 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}",1,"(-4*(a - b)*b*Sqrt[a + b]*(3*A*b^2 - a^2*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(25*a^2*A - 57*a*A*b - 6*A*b^2 + 35*a^2*C - 105*a*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (6*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,7,37,0.1892,1,"{4221, 3048, 3047, 3055, 2998, 2816, 2994}"
1421,1,525,0,1.3638377,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*Sqrt[a + b]*(A*b^2 + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A*b^2 - 2*a*b*(2*A + 5*C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,8,37,0.2162,1,"{4221, 3048, 3047, 3053, 2809, 2998, 2816, 2994}"
1422,1,560,0,1.6959941,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"((a - b)*b*Sqrt[a + b]*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A + 3*C) - a*(8*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - (b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,37,0.2432,1,"{4221, 3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1423,1,569,0,1.7288166,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}",1,"((a - b)*Sqrt[a + b]*(8*A - 5*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a*A - 16*A*b - 5*a*C - 2*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (b*(4*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - (a*(8*A - 5*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,9,37,0.2432,1,"{4221, 3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1424,1,613,0,1.9063278,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(48*a*A*b + 24*A*b^2 + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(24*A*b^2 - a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)","A",9,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1425,1,698,0,2.364233,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{64 b^2 d}-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 b C+3 a^3 C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(24 a^2 b^2 (2 A+C)+3 a^4 C+16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{8 b d \sqrt{\sec (c+d x)}}","\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{64 b^2 d}-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 b C+3 a^3 C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(24 a^2 b^2 (2 A+C)+3 a^4 C+16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{8 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^3*C - 2*a^2*b*C - 8*b^3*(4*A + 3*C) - 4*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^4*C + 24*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) - ((3*a^2*C - 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) - (a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*b^2*d)","A",10,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1426,1,627,0,2.6146168,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d}-\frac{2 \left(-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}","\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d}-\frac{2 \left(-a^2 b^2 (205 A+297 C)-15 a^4 (9 A+11 C)+4 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)+15 a^4 (9 A+11 C)+6 a A b^3+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}",1,"(2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*a^2*d) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(693*a*d) + (2*(5*A*b^2 + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",9,7,37,0.1892,1,"{4221, 3048, 3047, 3055, 2998, 2816, 2994}"
1427,1,544,0,2.0072255,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 A+28 C)+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)+10 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}","\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 b (19 A+28 C)+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)+10 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}",1,"(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 21*a^3*(7*A + 9*C) + 15*a*b^2*(11*A + 21*C) - 6*a^2*b*(19*A + 28*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*(5*A*b^2 + a^2*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,7,37,0.1892,1,"{4221, 3048, 3047, 3055, 2998, 2816, 2994}"
1428,1,600,0,1.8235519,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}-\frac{2 \sqrt{a+b} \left(a^2 b (29 A+49 C)+a^3 (-(5 A+7 C))-9 a b^2 (3 A+7 C)+3 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}-\frac{2 \sqrt{a+b} \left(a^2 b (29 A+49 C)+a^3 (-(5 A+7 C))-9 a b^2 (3 A+7 C)+3 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*b*Sqrt[a + b]*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(3*A*b^3 - 9*a*b^2*(3*A + 7*C) - a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,8,37,0.2162,1,"{4221, 3048, 3047, 3053, 2809, 2998, 2816, 2994}"
1429,1,666,0,2.3386798,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{\sqrt{a+b} \left(a^2 (34 A b+90 b C)-6 a^3 (3 A+5 C)-a b^2 (46 A-15 C)+30 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{5 a b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{\sqrt{a+b} \left(a^2 (34 A b+90 b C)-6 a^3 (3 A+5 C)-a b^2 (46 A-15 C)+30 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{5 a b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(30*A*b^3 - a*b^2*(46*A - 15*C) - 6*a^3*(3*A + 5*C) + a^2*(34*A*b + 90*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (5*a*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) - ((b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",10,9,37,0.2432,1,"{4221, 3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1430,1,627,0,2.2799326,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{10 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{10 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((a - b)*b*Sqrt[a + b]*(56*A - 27*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*b^2*(12*A + C) + 8*a^2*(A + 3*C) - a*(56*A*b - 27*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (b^2*(8*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - (a*b*(56*A - 27*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",10,10,37,0.2703,1,"{4221, 3048, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1431,1,669,0,2.4188925,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}",1,"((a - b)*Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C) - 2*a*b*(72*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (5*a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) - (a*b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(6*A - C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - ((a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",10,9,37,0.2432,1,"{4221, 3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1432,1,695,0,2.3232196,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (192 A+59 C)+15 a^3 C+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-120 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}","\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (192 A+59 C)+15 a^3 C+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-120 a^2 b^2 (2 A+C)+5 a^4 C-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C + 24*b^3*(4*A + 3*C) + 2*a^2*b*(192*A + 59*C) + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4*C - 120*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)","A",10,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1433,1,806,0,3.0961495,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}-\frac{\left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{a \left(-15 C a^2+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b C a^3-12 b^2 (220 A+141 C) a^2-8 b^3 (260 A+193 C) a-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(3 C a^4+40 b^2 (2 A+C) a^2+80 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}-\frac{\left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{a \left(-15 C a^2+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b C a^3-12 b^2 (220 A+141 C) a^2-8 b^3 (260 A+193 C) a-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(3 C a^4+40 b^2 (2 A+C) a^2+80 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) - (3*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) - ((45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)","A",11,9,37,0.2432,1,"{4221, 3050, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1434,1,469,0,1.4334761,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}-\frac{2 \sqrt{a+b} \left(-a^2 (44 A b+70 b C)-5 a^3 (5 A+7 C)+12 a A b^2-48 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}-\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}","\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}-\frac{2 \sqrt{a+b} \left(-a^2 (44 A b+70 b C)-5 a^3 (5 A+7 C)+12 a A b^2-48 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}-\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}",1,"(-4*(a - b)*b*Sqrt[a + b]*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(12*a*A*b^2 - 48*A*b^3 - 5*a^3*(5*A + 7*C) - a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^3*d) - (12*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*a*d)","A",7,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1435,1,394,0,1.0099005,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}-\frac{8 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}","\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}-\frac{8 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*a*A*b - 8*A*b^2 - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (8*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)","A",6,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1436,1,323,0,0.6793438,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}","\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(-4*A*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*A*b + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",5,5,37,0.1351,1,"{4221, 3056, 2998, 2816, 2994}"
1437,1,403,0,0.6196561,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 A \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 A \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*A*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3054, 2809, 12, 2801, 2816, 2994}"
1438,1,453,0,0.9145847,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}+\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}+\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}",1,"-(((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A*b + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",7,7,37,0.1892,1,"{4221, 3062, 3053, 2809, 2998, 2816, 2994}"
1439,1,515,0,1.2388566,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{C (3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{C (3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(3*(a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - ((3*a - 2*b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2*C + 4*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) - (3*a*C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)","A",8,8,37,0.2162,1,"{4221, 3050, 3061, 3053, 2809, 2998, 2816, 2994}"
1440,1,534,0,1.6999605,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}-\frac{2 \left(2 a^2 b (2 A+5 C)+a^3 (3 A+5 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \left(-2 a^2 b^2 (4 A-5 C)+a^4 (-(3 A+5 C))+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}-\frac{2 \left(2 a^2 b (2 A+5 C)+a^3 (3 A+5 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \left(-2 a^2 b^2 (4 A-5 C)+a^4 (-(3 A+5 C))+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*(16*A*b^4 - 2*a^2*b^2*(4*A - 5*C) - a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(12*a*A*b^2 + 16*A*b^3 + 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(8*A*b^2 - a^2*(3*A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)","A",7,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1441,1,432,0,1.1923468,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(2*b*(8*A*b^2 - a^2*(5*A - 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(6*a*A*b + 8*A*b^2 + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)","A",6,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1442,1,348,0,0.8161862,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 (a (A-C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 (a (A-C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*(2*A*b^2 - a^2*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b + a*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,37,0.1351,1,"{4221, 3056, 2998, 2816, 2994}"
1443,1,481,0,0.9972788,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 (A b-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 (A b-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}",1,"(2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,37,0.1892,1,"{4221, 3052, 2809, 2993, 2998, 2816, 2994}"
1444,1,563,0,1.4976573,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"-(((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + ((2*A*b^2 + a*(3*a + b)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (3*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",8,8,37,0.2162,1,"{4221, 3048, 3061, 3053, 2809, 2998, 2816, 2994}"
1445,1,664,0,1.9838161,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}",1,"((8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((8*A*b^2 + (15*a^2 + 5*a*b - 2*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A*b^2 + 5*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d)","A",9,9,37,0.2432,1,"{4221, 3048, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1446,1,589,0,1.8833699,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{4 \left(5 a^2 A b^2+2 a^4 C-3 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 b^2 (8 A-C)-a^3 (9 A b-3 b C)+a^4 (-(A+3 C))+12 a A b^3+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{4 b \left(-a^2 b^2 (14 A-C)+a^4 (4 A-3 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}","\frac{2 \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{4 \left(5 a^2 A b^2+2 a^4 C-3 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 b^2 (8 A-C)-a^3 (9 A b-3 b C)+a^4 (-(A+3 C))+12 a A b^3+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{4 b \left(-a^2 b^2 (14 A-C)+a^4 (4 A-3 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(-4*b*(8*A*b^4 + a^4*(4*A - 3*C) - a^2*b^2*(14*A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) - (2*(12*a*A*b^3 + 16*A*b^4 - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C) - a^3*(9*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",7,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1447,1,489,0,1.3469451,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{4 \left(-a^2 b^2 (4 A+C)+a^4 (-C)+2 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b (9 A+C)-3 a^3 (A-C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}","-\frac{4 \left(-a^2 b^2 (4 A+C)+a^4 (-C)+2 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-a^2 b (9 A+C)-3 a^3 (A-C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(2*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(6*a*A*b^2 + 8*A*b^3 - 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*(2*A*b^4 - a^4*C - a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3056, 3055, 2998, 2816, 2994}"
1448,1,456,0,1.2405475,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 (-(3 A+C))+3 a b (A+C)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 (-(3 A+C))+3 a b (A+C)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-4*b*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b^2 + 3*a*b*(A + C) - a^2*(3*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (4*b*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,37,0.1622,1,"{4221, 3056, 2993, 2998, 2816, 2994}"
1449,1,618,0,1.7835625,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 b C+3 a^3 C-3 a b^2 (A+2 C)+A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 b C+3 a^3 C-3 a b^2 (A+2 C)+A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 \left(a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"(-2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^3 + 3*a^3*C + a^2*b*C - 3*a*b^2*(A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,37,0.2162,1,"{4221, 3048, 3051, 2809, 2993, 2998, 2816, 2994}"
1450,1,710,0,2.4281986,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(21 a^2 b^2 C-5 a^3 b C-15 a^4 C-a b^3 (2 A-3 C)+6 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{5 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d \sqrt{\sec (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(21 a^2 b^2 C-5 a^3 b C-15 a^4 C-a b^3 (2 A-3 C)+6 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left(8 A b^4-C \left(-26 a^2 b^2+15 a^4+3 b^4\right)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{5 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d \sqrt{\sec (c+d x)}}",1,"((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - ((6*A*b^4 - a*b^3*(2*A - 3*C) - 15*a^4*C - 5*a^3*b*C + 21*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (5*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)","A",9,9,37,0.2432,1,"{4221, 3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1451,1,230,0,0.3762888,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(-2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,7,41,0.1707,1,"{4221, 3031, 3021, 2748, 2636, 2641, 2639}"
1452,1,192,0,0.3460596,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*(3*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a*A + 5*b*B + 5*a*C)*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*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,41,0.1707,1,"{4221, 3031, 3021, 2748, 2636, 2639, 2641}"
1453,1,151,0,0.3204464,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*(A*b + a*B - b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*b*B + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*b + a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4221, 3031, 3021, 2748, 2641, 2639}"
1454,1,147,0,0.3015952,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*(b*B - a*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A*b + 3*a*B + b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,41,0.1463,1,"{4221, 3031, 3023, 2748, 2641, 2639}"
1455,1,156,0,0.2958739,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(b*B + a*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,6,41,0.1463,1,"{4221, 3033, 3023, 2748, 2641, 2639}"
1456,1,194,0,0.3128744,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b + 7*a*B + 5*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",7,7,41,0.1707,1,"{4221, 3033, 3023, 2748, 2639, 2635, 2641}"
1457,1,230,0,0.3586986,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 \sin (c+d x) (9 a B+9 A b+7 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (7 a A+5 a C+5 b B)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 \sin (c+d x) (9 a B+9 A b+7 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (7 a A+5 a C+5 b B)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(9*A*b + 9*a*B + 7*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(7*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4221, 3033, 3023, 2748, 2635, 2641, 2639}"
1458,1,342,0,0.7634208,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(-2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*a*(4*A*b + 9*a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,8,43,0.1860,1,"{4221, 3047, 3031, 3021, 2748, 2636, 2641, 2639}"
1459,1,288,0,0.6981357,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(-2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,8,43,0.1860,1,"{4221, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1460,1,240,0,0.651969,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 a (5 a B+4 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 a (5 a B+4 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(-2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(4*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(4*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,43,0.1628,1,"{4221, 3047, 3031, 3021, 2748, 2641, 2639}"
1461,1,220,0,0.6197876,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*(4*A*b + 3*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,43,0.1628,1,"{4221, 3047, 3031, 3023, 2748, 2641, 2639}"
1462,1,229,0,0.6492284,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) (6 a A-2 a C-b B)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) (6 a A-2 a C-b B)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(5*A - C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a*A - b*B - 2*a*C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,7,43,0.1628,1,"{4221, 3047, 3033, 3023, 2748, 2641, 2639}"
1463,1,243,0,0.6351747,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}",1,"(2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3049, 3033, 3023, 2748, 2641, 2639}"
1464,1,291,0,0.6860443,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(9*b*B + 4*a*C)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,8,43,0.1860,1,"{4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1465,1,345,0,0.7490513,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 \sin (c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(11*b*B + 4*a*C)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(11*A*b^2 + 22*a*b*B + 4*a^2*C + 9*b^2*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4221, 3049, 3033, 3023, 2748, 2635, 2641, 2639}"
1466,1,397,0,1.0898955,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right)}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right)}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}",1,"(-2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,8,43,0.1860,1,"{4221, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1467,1,334,0,0.9862965,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(21 a^2 b (3 A+5 C)+21 a^3 B+98 a b^2 B+24 A b^3\right)}{35 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 (7 a B+6 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(21 a^2 b (3 A+5 C)+21 a^3 B+98 a b^2 B+24 A b^3\right)}{35 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 (7 a B+6 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}",1,"(-2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(24*A*b^3 + 21*a^3*B + 98*a*b^2*B + 21*a^2*b*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",8,7,43,0.1628,1,"{4221, 3047, 3031, 3021, 2748, 2641, 2639}"
1468,1,313,0,0.9625215,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"(-2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(24*A*b^2 + 35*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,7,43,0.1628,1,"{4221, 3047, 3031, 3023, 2748, 2641, 2639}"
1469,1,311,0,1.0019033,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{2 b \sin (c+d x) \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (15 a B+35 A b-3 b C)}{15 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}","-\frac{2 b \sin (c+d x) \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (15 a B+35 A b-3 b C)}{15 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"(-2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(35*A*b + 15*a*B - 3*b*C)*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a^2*B - b^2*B + 3*a*b*(5*A - C))*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,43,0.1628,1,"{4221, 3047, 3033, 3023, 2748, 2641, 2639}"
1470,1,319,0,0.9942467,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 b \sin (c+d x) \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (35 a A-11 a C-7 b B)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}","\frac{2 b \sin (c+d x) \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (35 a A-11 a C-7 b B)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}",1,"(2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(35*a*A - 7*b*B - 11*a*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a*b*B - 6*a^2*(7*A - 3*C) + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(7*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,43,0.1860,1,"{4221, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1471,1,336,0,1.0073635,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 b \sin (c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}","\frac{2 b \sin (c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}",1,"(2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (2*(3*b*B + 2*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3049, 3033, 3023, 2748, 2641, 2639}"
1472,1,401,0,1.0606136,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \left(242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Sin[c + d*x])/(495*d*Sec[c + d*x]^(3/2)) + (2*(11*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1473,1,463,0,1.141254,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 \sin (c+d x) \left(39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(338 a^2 b B+24 a^3 C+39 a b^2 (11 A+9 C)+117 b^3 B\right)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{195 d}+\frac{2 (6 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(338 a^2 b B+24 a^3 C+39 a b^2 (11 A+9 C)+117 b^3 B\right)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^2 b (9 A+7 C)+117 a^3 B+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{195 d}+\frac{2 (6 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(143*A*b^2 + 195*a*b*B + 24*a^2*C + 121*b^2*C)*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(338*a^2*b*B + 117*b^3*B + 24*a^3*C + 39*a*b^2*(11*A + 9*C))*Sin[c + d*x])/(1001*d*Sec[c + d*x]^(5/2)) + (2*(13*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,8,43,0.1860,1,"{4221, 3049, 3033, 3023, 2748, 2635, 2641, 2639}"
1474,1,515,0,1.5902503,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right)}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right)}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}",1,"(-2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",10,8,43,0.1860,1,"{4221, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1475,1,441,0,1.4922815,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+756 a^3 b B+1098 a b^3 B+192 A b^4\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+756 a^3 b B+1098 a b^3 B+192 A b^4\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}",1,"(-2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(192*A*b^4 + 756*a^3*b*B + 1098*a*b^3*B + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",9,7,43,0.1628,1,"{4221, 3047, 3031, 3021, 2748, 2641, 2639}"
1476,1,423,0,1.4685866,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right)}{105 d}-\frac{2 b^2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right)}{105 d}-\frac{2 b^2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"(-2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(192*A*b^3 + 63*a^3*B + 413*a*b^2*B + a^2*(202*A*b + 350*b*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,7,43,0.1628,1,"{4221, 3047, 3031, 3023, 2748, 2641, 2639}"
1477,1,426,0,1.4717591,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 b^2 \sin (c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}","-\frac{2 b^2 \sin (c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(-2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(50*a*b*B + b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*b*(105*a^2*b*B - 5*b^3*B + 4*a*b^2*(33*A - 5*C) + 6*a^3*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + 15*a*b*B + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,7,43,0.1628,1,"{4221, 3047, 3033, 3023, 2748, 2641, 2639}"
1478,1,413,0,1.4510106,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{2 b^2 \sin (c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left(3 a^2 b (49 A-13 C)+42 a^3 B-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b \sin (c+d x) (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+8 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}","-\frac{2 b^2 \sin (c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left(3 a^2 b (49 A-13 C)+42 a^3 B-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b \sin (c+d x) (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+8 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}",1,"(-2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) - (2*b*(42*a^3*B - 28*a*b^2*B + 3*a^2*b*(49*A - 13*C) - b^3*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*A*b + 7*a*B - b*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,8,43,0.1860,1,"{4221, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1479,1,419,0,1.4444493,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 b^2 \sin (c+d x) \left(a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}-\frac{2 b \sin (c+d x) (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}","\frac{2 b^2 \sin (c+d x) \left(a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}-\frac{2 b \sin (c+d x) (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}",1,"(2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(162*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(117*a^2*b*B + 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*a*A - 3*b*B - 5*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,8,43,0.1860,1,"{4221, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1480,1,444,0,1.4601075,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{2 b \sin (c+d x) \left(1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}","\frac{2 b \sin (c+d x) \left(1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}",1,"(2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(11*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])","A",9,7,43,0.1628,1,"{4221, 3049, 3033, 3023, 2748, 2641, 2639}"
1481,1,517,0,1.5274192,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \left(11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(3/2)) + (2*(13*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,8,43,0.1860,1,"{4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1482,1,294,0,1.4220002,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}",1,"(-2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (2*b*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) - (2*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)","A",9,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1483,1,218,0,0.9809583,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^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)}{a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^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)}{a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",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*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",8,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1484,1,178,0,0.6674042,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",7,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1485,1,157,0,0.4345135,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)","A",6,6,43,0.1395,1,"{4221, 3059, 2639, 3002, 2641, 2805}"
1486,1,207,0,0.7539487,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"(2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(b^2*(3*A + C) - 3*a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3049, 3059, 2639, 3002, 2641, 2805}"
1487,1,270,0,1.0587154,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) + (2*(3*a^2*b*B + b^3*B - 3*a^3*C - a*b^2*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^4*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Sec[c + d*x]^(3/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3049, 3059, 2639, 3002, 2641, 2805}"
1488,1,345,0,1.4780977,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{2 \sin (c+d x) \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (3 A+C)+21 a^3 b B-21 a^4 C+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 b B-5 a^3 C-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (3 A+C)+21 a^3 b B-21 a^4 C+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 b B-5 a^3 C-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*b^5*d) - (2*a^3*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^5*(a + b)*d) + (2*C*Sin[c + d*x])/(7*b*d*Sec[c + d*x]^(5/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(5*b^2*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*b^3*d*Sqrt[Sec[c + d*x]])","A",9,7,43,0.1628,1,"{4221, 3049, 3059, 2639, 3002, 2641, 2805}"
1489,1,452,0,1.7362408,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 C-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 C-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"-(((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1490,1,366,0,1.2737441,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1491,1,303,0,0.8622561,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 C+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 C+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"-(((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1492,1,311,0,0.8806769,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b B-3 a^3 C+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b B-3 a^3 C+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + a*b^2*(A + 4*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",7,7,43,0.1628,1,"{4221, 3047, 3059, 2639, 3002, 2641, 2805}"
1493,1,403,0,1.3254525,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\sin (c+d x) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-16 C)+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 C-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}","\frac{\sin (c+d x) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-16 C)+9 a^3 b B-15 a^4 C-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 C-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - a^2*b^2*(3*A - 16*C) - 15*a^4*C + 2*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",8,8,43,0.1860,1,"{4221, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1494,1,505,0,1.8057434,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\sin (c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(5 a^2 b B-7 a^3 C-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+15 a^4 b B-21 a^5 C+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^2 b^2 (5 A-8 C)+25 a^3 b B-35 a^4 C-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^2 b^2 (A-3 C)+5 a^3 b B-7 a^4 C-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}","\frac{\sin (c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(5 a^2 b B-7 a^3 C-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+15 a^4 b B-21 a^5 C+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^2 b^2 (5 A-8 C)+25 a^3 b B-35 a^4 C-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^2 b^2 (A-3 C)+5 a^3 b B-7 a^4 C-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}",1,"-((25*a^3*b*B - 20*a*b^3*B - 3*a^2*b^2*(5*A - 8*C) - 35*a^4*C + 2*b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - a^3*b^2*(9*A - 20*C) - 21*a^5*C + 4*a*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 + 5*a^3*b*B - 7*a*b^3*B - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + ((5*a^2*b*B - 2*b^3*B - a*b^2*(3*A - 4*C) - 7*a^3*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4221, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1495,1,669,0,2.8158576,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",10,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1496,1,562,0,2.0898183,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 C-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 C-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"-((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1497,1,473,0,1.5081659,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","-\frac{\sin (c+d x) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}","-\frac{\sin (c+d x) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 C+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3055, 3059, 2639, 3002, 2641, 2805}"
1498,1,478,0,1.5817074,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","-\frac{\sin (c+d x) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (3 A-5 C)+a^3 b B+3 a^4 C-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}","-\frac{\sin (c+d x) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (3 A-5 C)+a^3 b B+3 a^4 C-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 C+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,8,43,0.1860,1,"{4221, 3047, 3055, 3059, 2639, 3002, 2641, 2805}"
1499,1,483,0,1.6832287,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 C-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 b^2 (A+33 C)-5 a^2 b^3 B+3 a^4 b B-15 a^5 C-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 C-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 C-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 C-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 b^2 (A+33 C)-5 a^2 b^3 B+3 a^4 b B-15 a^5 C-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 C-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 C-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"-((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C - a*b^4*(7*A + 24*C) + a^3*b^2*(A + 33*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4221, 3047, 3059, 2639, 3002, 2641, 2805}"
1500,1,596,0,2.1297904,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","-\frac{\sin (c+d x) \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-99 a^3 b^3 B+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}","-\frac{\sin (c+d x) \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-99 a^3 b^3 B+45 a^5 b B-105 a^6 C+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - a^4*b^2*(9*A - 223*C) + a^2*b^4*(15*A - 128*C) - 105*a^6*C - 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4221, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1501,1,714,0,3.0484894,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","-\frac{\sin (c+d x) \left(-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}","-\frac{\sin (c+d x) \left(-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}",1,"-((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d) + ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C) - 189*a^7*C - 24*a*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + ((35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",10,8,43,0.1860,1,"{4221, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1502,1,592,0,2.2472131,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b (13 A+21 C)+75 a^3 B-12 a b^2 B+8 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)+12 a b^2 (A-2 B)+16 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-24 a b^3 B+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b (13 A+21 C)+75 a^3 B-12 a b^2 B+8 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)+12 a b^2 (A-2 B)+16 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-24 a b^3 B+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}",1,"(-2*(a - b)*Sqrt[a + b]*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^5*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(16*A*b^3 + 12*a*b^2*(A - 2*B) + 6*a^2*b*(6*A - 3*B + 7*C) + 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a^2*d) + (2*(A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1503,1,487,0,1.5889129,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 2*a*b*(3*A - 7*B) + a^2*(25*A - 63*B + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1504,1,400,0,1.1012344,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}","-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(2*A*b + a*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1505,1,467,0,1.0063968,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(b*(A - 3*B) - a*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,45,0.1556,1,"{4221, 3047, 3053, 2809, 2998, 2816, 2994}"
1506,1,509,0,1.3208971,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(2*A - C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*A*b - a*(2*A - 2*B - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,45,0.1778,1,"{4221, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1507,1,543,0,1.3533457,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*A*b + a*C + 2*b*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)","A",8,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1508,1,646,0,1.9794605,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b B+a^3 (-C)-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b B+a^3 (-C)-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(24*A*b^2 + (a + 2*b)*(6*b*B - 3*a*C + 8*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) + ((8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)","A",9,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1509,1,766,0,2.6566654,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\sin (c+d x) \left(5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (12 B+5 C)+15 a^3 C+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-8 a^2 b^2 (2 A+C)+8 a^3 b B-5 a^4 C+32 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\sin (c+d x) \left(5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (12 B+5 C)+15 a^3 C+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-8 a^2 b^2 (2 A+C)+8 a^3 b B-5 a^4 C+32 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"((a - b)*Sqrt[a + b]*(24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 2*a^2*b*(12*B + 5*C) + 4*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 16*B + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a^3*b*B + 32*a*b^3*B - 5*a^4*C - 8*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((16*A*b^2 - 8*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)","A",10,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1510,1,590,0,2.2187819,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1511,1,490,0,1.5676445,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(25 A-63 B+35 C))+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (41 A+70 C)-63 a^3 B-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(25 A-63 B+35 C))+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (41 A+70 C)-63 a^3 B-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}",1,"(-2*(a - b)*Sqrt[a + b]*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B + 35*C) + 3*a*b*(19*A - 7*B + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1512,1,550,0,1.4301436,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*Sqrt[a + b]*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(3*b^2*(A - 5*B) - 2*a*b*(6*A - 10*B + 15*C) + a^2*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,7,45,0.1556,1,"{4221, 3047, 3053, 2809, 2998, 2816, 2994}"
1513,1,588,0,1.8672174,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{\sqrt{a+b} (3 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{\sqrt{a+b} (3 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(8*A*b + 6*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A - 3*B + 3*C) - a*b*(8*A - 3*(4*B + C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((8*A*b + 6*a*B - 3*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,8,45,0.1778,1,"{4221, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1514,1,595,0,1.8890182,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d \sqrt{\sec (c+d x)}}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d \sqrt{\sec (c+d x)}}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}",1,"((a - b)*Sqrt[a + b]*(8*a*A - 4*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a*(8*A - 8*B - 5*C) - 2*b*(8*A + 2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (b*(4*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((8*a*A - 4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,9,45,0.2000,1,"{4221, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1515,1,647,0,1.980921,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b B+a^3 (-C)+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(3*a^2*C + 4*b^2*(6*A + 3*B + 4*C) + 2*a*b*(24*A + 15*B + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)","A",9,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1516,1,764,0,2.7690693,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4*a*b^2*(60*A + 28*B + 39*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) + ((4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d)","A",10,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1517,1,705,0,3.3718297,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 b^2 (19 A-121 B+33 C)-6 a^3 b (505 A-209 B+660 C)+3 a^4 (225 A-539 B+275 C)+10 a b^3 (3 A-11 B)+40 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}","\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 b^2 (19 A-121 B+33 C)-6 a^3 b (505 A-209 B+660 C)+3 a^4 (225 A-539 B+275 C)+10 a b^3 (3 A-11 B)+40 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}",1,"(2*(a - b)*Sqrt[a + b]*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 10*a*b^3*(3*A - 11*B) + 15*a^2*b^2*(19*A - 121*B + 33*C) + 3*a^4*(225*A - 539*B + 275*C) - 6*a^3*b*(505*A - 209*B + 660*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",9,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1518,1,592,0,2.2839315,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+5 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b (163 A+231 C)+75 a^3 B+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b (19 A-60 B+28 C)+3 a^3 (49 A-25 B+63 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+5 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}",1,"(-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,45,0.1333,1,"{4221, 3047, 3055, 2998, 2816, 2994}"
1519,1,640,0,1.9939924,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 b (145 A-119 B+245 C)+a^3 (-(25 A-63 B+35 C))-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(5 a^2 b (29 A+49 C)+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+5 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 b (145 A-119 B+245 C)+a^3 (-(25 A-63 B+35 C))-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(5 a^2 b (29 A+49 C)+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+5 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*Sqrt[a + b]*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(15*b^3*(A - 7*B) - a^3*(25*A - 63*B + 35*C) + a^2*b*(145*A - 119*B + 245*C) - a*b^2*(135*A - 161*B + 315*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(5*A*b + 7*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,7,45,0.1556,1,"{4221, 3047, 3053, 2809, 2998, 2816, 2994}"
1520,1,703,0,2.5562557,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (17 A-35 B+45 C)-2 a^3 (9 A-5 B+15 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}-\frac{b \sqrt{a+b} (5 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (17 A-35 B+45 C)-2 a^3 (9 A-5 B+15 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}-\frac{b \sqrt{a+b} (5 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(30*A*b^3 - 2*a^3*(9*A - 5*B + 15*C) + 2*a^2*b*(17*A - 35*B + 45*C) - a*b^2*(46*A - 15*(6*B + C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) - ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(A*b + a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",10,8,45,0.1778,1,"{4221, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1521,1,682,0,2.428319,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}",1,"((a - b)*Sqrt[a + b]*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a*b*(56*A - 72*B - 27*C) - 6*b^2*(12*A + 2*B + C) - 8*a^2*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(8*A*b + 4*a*B - b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",10,9,45,0.2000,1,"{4221, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1522,1,707,0,2.590199,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}",1,"-((a - b)*Sqrt[a + b]*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a^2*(48*A - 48*B - 33*C) - 4*b^2*(6*A + 3*B + 4*C) - 2*a*b*(72*A + 27*B + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) - (b*(8*a*A - 2*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(6*A - C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",10,9,45,0.2000,1,"{4221, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1523,1,760,0,2.802975,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sin (c+d x) \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sin (c+d x) \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(12*A + 16*B + 9*C) + 2*a^2*b*(192*A + 132*B + 59*C) + 4*a*b^2*(108*A + 52*B + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + ((8*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)","A",10,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1524,1,894,0,4.0581259,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(260*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) + ((10*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)","A",11,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1525,1,506,0,1.6077978,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}+\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (22 A+7 (B+5 C))+a^3 (25 A-63 B+35 C)-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (44 A b+70 b C)-63 a^3 B-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}-\frac{2 (6 A b-7 a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}+\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 b (22 A+7 (B+5 C))+a^3 (25 A-63 B+35 C)-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (44 A b+70 b C)-63 a^3 B-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}-\frac{2 (6 A b-7 a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}",1,"(-2*(a - b)*Sqrt[a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2*(3*A + 14*B) + a^3*(25*A - 63*B + 35*C) + 2*a^2*b*(22*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^3*d) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*a*d)","A",7,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1526,1,412,0,1.0938314,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (4 A b-5 a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}","-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (4 A b-5 a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}",1,"(2*(a - b)*Sqrt[a + b]*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(8*A*b^2 - 2*a*b*(A + 5*B) + a^2*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)","A",6,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1527,1,333,0,0.7122952,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}","\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(-2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",5,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1528,1,407,0,0.6808267,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A-B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A-B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A - B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3053, 2809, 2998, 2816, 2994}"
1529,1,461,0,0.9284676,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}",1,"-(((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A*b + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",7,7,45,0.1556,1,"{4221, 3061, 3053, 2809, 2998, 2816, 2994}"
1530,1,545,0,1.2975291,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)","A",8,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1531,1,653,0,2.0862074,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b B-5 a^3 C-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d \sqrt{\sec (c+d x)}}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b B-5 a^3 C-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d \sqrt{\sec (c+d x)}}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"-((a - b)*Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 12*b^2*B + 15*a^2*C - 10*a*b*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d*Sec[c + d*x]^(3/2)) + ((6*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^3*d)","A",9,8,45,0.1778,1,"{4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1532,1,445,0,1.2544234,"\int \frac{\left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} (2 A+B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (a B+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{B (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} (2 A+B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (a B+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{B (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}",1,"-(((a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A + B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b + a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,54,0.1481,1,"{4221, 3029, 3003, 3053, 2809, 2998, 2816, 2994}"
1533,1,585,0,1.9554883,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 (-(A-5 C))-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right)}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 (-(A-5 C))-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right)}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^5*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(48*A*b^3 + 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)","A",7,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1534,1,464,0,1.2825203,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 (5 A b-3 b C)+3 a^3 B-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(A-3 C))-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 (5 A b-3 b C)+3 a^3 B-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^2 + 6*a*b*(A - B) + a^2*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)","A",6,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1535,1,362,0,0.8423772,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*(2*A*b^2 - a*b*B - a^2*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b + a*(A - B - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1536,1,496,0,1.0693046,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}",1,"(2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b + b*B - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,45,0.1556,1,"{4221, 3051, 2809, 2993, 2998, 2816, 2994}"
1537,1,595,0,1.6857345,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"-(((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + ((2*A*b^2 - a*(b*(2*B - C) - 3*a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",8,8,45,0.1778,1,"{4221, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
1538,1,720,0,2.3454231,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}",1,"-((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((8*A*b^2 - a*b*(12*B - 5*C) + 15*a^2*C - 2*b^2*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d)","A",9,9,45,0.2000,1,"{4221, 3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
1539,1,660,0,2.7341548,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(10 a^2 A b^2-7 a^3 b B+4 a^4 C+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b^2 (8 A+3 B-C)-3 a^3 b (3 A-3 B-C)+a^4 (-(A-3 B+3 C))+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(10 a^2 A b^2-7 a^3 b B+4 a^4 C+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b^2 (8 A+3 B-C)-3 a^3 b (3 A-3 B-C)+a^4 (-(A-3 B+3 C))+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(-2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) - (2*(16*A*b^4 + 4*a*b^3*(3*A - 2*B) - 3*a^3*b*(3*A - 3*B - C) - 2*a^2*b^2*(8*A + 3*B - C) - a^4*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",7,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1540,1,535,0,1.6460626,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 C-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 b (9 A+3 B+C)-3 a^3 (A-B-C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 C-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 b (9 A+3 B+C)-3 a^3 (A-B-C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 + 2*a*b^2*(3*A - B) - 3*a^3*(A - B - C) - a^2*b*(9*A + 3*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,5,45,0.1111,1,"{4221, 3055, 2998, 2816, 2994}"
1541,1,495,0,1.3763679,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(3 A+3 B+C))+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-(3 A+3 B+C))+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^2 b (3 A+2 C)+3 a^3 B+a b^2 B+2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b^2 - a^2*(3*A + 3*B + C) + a*b*(3*A + B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,45,0.1333,1,"{4221, 3055, 2993, 2998, 2816, 2994}"