1,1,125,0,0.1670908,"\int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","-\frac{a (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (A+B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d}","-\frac{a (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (A+B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(3*a*(A + B)*x)/8 + (a*(5*A + 4*B)*Sin[c + d*x])/(5*d) + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 4*B)*Sin[c + d*x]^3)/(15*d)","A",8,6,29,0.2069,1,"{2968, 3023, 2748, 2633, 2635, 8}"
2,1,97,0,0.1490146,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","-\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d}","-\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(4*A + 3*B)*x)/8 + (a*(A + B)*Sin[c + d*x])/d + (a*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)","A",7,6,29,0.2069,1,"{2968, 3023, 2748, 2635, 8, 2633}"
3,1,77,0,0.0783281,"\int \cos (c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d}","\frac{a (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(A + B)*x)/2 + (a*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",3,3,27,0.1111,1,"{2968, 3023, 2734}"
4,1,47,0,0.020611,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (2 A+B)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (2 A+B)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*A + B)*x)/2 + (a*(A + B)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,21,0.04762,1,"{2734}"
5,1,32,0,0.0913122,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","a x (A+B)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x)}{d}","a x (A+B)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x)}{d}",1,"a*(A + B)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d","A",4,4,27,0.1481,1,"{2968, 3023, 2735, 3770}"
6,1,32,0,0.1034911,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*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 B x","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+a B x",1,"a*B*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d","A",4,4,29,0.1379,1,"{2968, 3021, 2735, 3770}"
7,1,56,0,0.1369258,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (A+2 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (A+2 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*(A + 2*B)*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",6,6,29,0.2069,1,"{2968, 3021, 2748, 3767, 8, 3770}"
8,1,86,0,0.1536549,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a (2 A+3 B) \tan (c+d x)}{3 d}+\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{a (2 A+3 B) \tan (c+d x)}{3 d}+\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*B)*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",7,7,29,0.2414,1,"{2968, 3021, 2748, 3768, 3770, 3767, 8}"
9,1,106,0,0.1646549,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a (A+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 B) \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+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(A + B)*Tan[c + d*x])/d + (a*(3*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(A + B)*Tan[c + d*x]^3)/(3*d)","A",7,6,29,0.2069,1,"{2968, 3021, 2748, 3767, 3768, 3770}"
10,1,191,0,0.3104454,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","-\frac{a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^4(c+d x)}{30 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (12 A+11 B)+\frac{B \sin (c+d x) \cos ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}","-\frac{a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^4(c+d x)}{30 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (12 A+11 B)+\frac{B \sin (c+d x) \cos ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}",1,"(a^2*(12*A + 11*B)*x)/16 + (a^2*(9*A + 8*B)*Sin[c + d*x])/(5*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*(6*A + 7*B)*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (B*Cos[c + d*x]^4*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a^2*(9*A + 8*B)*Sin[c + d*x]^3)/(15*d)","A",9,7,31,0.2258,1,"{2976, 2968, 3023, 2748, 2633, 2635, 8}"
11,1,160,0,0.2778279,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","-\frac{a^2 (10 A+9 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (5 A+6 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+6 B)+\frac{B \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 A+9 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (5 A+6 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+6 B)+\frac{B \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*A + 6*B)*x)/8 + (a^2*(10*A + 9*B)*Sin[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*A + 6*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(10*A + 9*B)*Sin[c + d*x]^3)/(15*d)","A",8,7,31,0.2258,1,"{2976, 2968, 3023, 2748, 2635, 8, 2633}"
12,1,129,0,0.176065,"\int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B)+\frac{(4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}","\frac{a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B)+\frac{(4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(8*A + 7*B)*x)/8 + (a^2*(8*A + 7*B)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A - B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)","A",4,4,29,0.1379,1,"{2968, 3023, 2751, 2644}"
13,1,94,0,0.0586307,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 A+2 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 A+2 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(3*A + 2*B)*x)/2 + (2*a^2*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a^2*(3*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",2,2,23,0.08696,1,"{2751, 2644}"
14,1,82,0,0.1926931,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}","\frac{a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}",1,"(a^2*(4*A + 3*B)*x)/2 + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B)*Sin[c + d*x])/(2*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d)","A",5,5,29,0.1724,1,"{2976, 2968, 3023, 2735, 3770}"
15,1,74,0,0.2096419,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{a^2 (A-B) \sin (c+d x)}{d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (A+2 B)+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}","-\frac{a^2 (A-B) \sin (c+d x)}{d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (A+2 B)+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}",1,"a^2*(A + 2*B)*x + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(A - B)*Sin[c + d*x])/d + (A*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d","A",5,5,31,0.1613,1,"{2975, 2968, 3023, 2735, 3770}"
16,1,88,0,0.2190875,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a^2 (3 A+2 B) \tan (c+d x)}{2 d}+\frac{a^2 (3 A+4 B) \tanh ^{-1}(\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 d}+a^2 B x","\frac{a^2 (3 A+2 B) \tan (c+d x)}{2 d}+\frac{a^2 (3 A+4 B) \tanh ^{-1}(\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 d}+a^2 B x",1,"a^2*B*x + (a^2*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*A + 2*B)*Tan[c + d*x])/(2*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,31,0.1613,1,"{2975, 2968, 3021, 2735, 3770}"
17,1,113,0,0.2701424,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a^2 (5 A+6 B) \tan (c+d x)}{3 d}+\frac{a^2 (2 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \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 A+6 B) \tan (c+d x)}{3 d}+\frac{a^2 (2 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \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*A + 3*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(5*A + 6*B)*Tan[c + d*x])/(3*d) + (a^2*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,7,31,0.2258,1,"{2975, 2968, 3021, 2748, 3767, 8, 3770}"
18,1,144,0,0.3036195,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a^2 (4 A+5 B) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 A+4 B) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \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 A+5 B) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 A+4 B) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \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*A + 8*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*A + 5*B)*Tan[c + d*x])/(3*d) + (a^2*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*A + 4*B)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,8,31,0.2581,1,"{2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
19,1,201,0,0.4321228,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","-\frac{a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac{a^3 (22 A+21 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \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) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}","-\frac{a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac{a^3 (22 A+21 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \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) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^3*(26*A + 23*B)*x)/16 + (a^3*(19*A + 17*B)*Sin[c + d*x])/(5*d) + (a^3*(26*A + 23*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(22*A + 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*A + 4*B)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(19*A + 17*B)*Sin[c + d*x]^3)/(15*d)","A",9,7,31,0.2258,1,"{2976, 2968, 3023, 2748, 2635, 8, 2633}"
20,1,154,0,0.2292039,"\int \cos (c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","-\frac{a^3 (15 A+13 B) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 A+13 B) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 A+13 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 A+13 B)+\frac{(5 A-B) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}","-\frac{a^3 (15 A+13 B) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 A+13 B) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 A+13 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 A+13 B)+\frac{(5 A-B) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(15*A + 13*B)*x)/8 + (a^3*(15*A + 13*B)*Sin[c + d*x])/(5*d) + (3*a^3*(15*A + 13*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*A - B)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(15*A + 13*B)*Sin[c + d*x]^3)/(60*d)","A",10,8,29,0.2759,1,"{2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
21,1,116,0,0.0984662,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","-\frac{a^3 (4 A+3 B) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \sin (c+d x)}{d}+\frac{3 a^3 (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 A+3 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","-\frac{a^3 (4 A+3 B) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \sin (c+d x)}{d}+\frac{3 a^3 (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 A+3 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(5*a^3*(4*A + 3*B)*x)/8 + (a^3*(4*A + 3*B)*Sin[c + d*x])/d + (3*a^3*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(4*A + 3*B)*Sin[c + d*x]^3)/(12*d)","A",8,6,23,0.2609,1,"{2751, 2645, 2637, 2635, 8, 2633}"
22,1,111,0,0.30435,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(3 A+5 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{1}{2} a^3 x (7 A+5 B)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(3 A+5 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{1}{2} a^3 x (7 A+5 B)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^3*(7*A + 5*B)*x)/2 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d)","A",6,5,29,0.1724,1,"{2976, 2968, 3023, 2735, 3770}"
23,1,110,0,0.3082044,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 A-B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 A+7 B)+\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}","\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 A-B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 A+7 B)+\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^3*(6*A + 7*B)*x)/2 + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*B*Sin[c + d*x])/(2*d) - ((2*A - B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d","A",6,6,31,0.1935,1,"{2975, 2976, 2968, 3023, 2735, 3770}"
24,1,114,0,0.337916,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a^3 (7 A+6 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+B) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}+a^3 x (A+3 B)-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}","\frac{a^3 (7 A+6 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+B) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}+a^3 x (A+3 B)-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"a^3*(A + 3*B)*x + (a^3*(7*A + 6*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + ((2*A + B)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/d + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,31,0.1613,1,"{2975, 2968, 3023, 2735, 3770}"
25,1,125,0,0.3366528,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac{a^3 (5 A+7 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 B x+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}","\frac{5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac{a^3 (5 A+7 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 B x+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"a^3*B*x + (a^3*(5*A + 7*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A + B)*Tan[c + d*x])/(2*d) + ((5*A + 3*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,5,31,0.1613,1,"{2975, 2968, 3021, 2735, 3770}"
26,1,154,0,0.4181484,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a^3 (9 A+11 B) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 A+28 B) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}","\frac{a^3 (9 A+11 B) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 A+28 B) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"(5*a^3*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(9*A + 11*B)*Tan[c + d*x])/(3*d) + (a^3*(27*A + 28*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A + 2*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",8,7,31,0.2258,1,"{2975, 2968, 3021, 2748, 3767, 8, 3770}"
27,1,185,0,0.44741,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a^3 (38 A+45 B) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 A+45 B) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 A+15 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}","\frac{a^3 (38 A+45 B) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 A+45 B) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 A+15 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^3*(13*A + 15*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 45*B)*Tan[c + d*x])/(15*d) + (a^3*(13*A + 15*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*A + 45*B)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((7*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,8,31,0.2581,1,"{2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
28,1,241,0,0.5927424,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","-\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{15 d}+\frac{a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^4 x (49 A+44 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}","-\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{15 d}+\frac{a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^4 x (49 A+44 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^4*(49*A + 44*B)*x)/16 + (a^4*(252*A + 227*B)*Sin[c + d*x])/(35*d) + (a^4*(49*A + 44*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(301*A + 276*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((7*A + 10*B)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (7*(A + B)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(252*A + 227*B)*Sin[c + d*x]^3)/(105*d)","A",10,7,31,0.2258,1,"{2976, 2968, 3023, 2748, 2635, 8, 2633}"
29,1,185,0,0.3039415,"\int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","-\frac{2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac{a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (8 A+7 B)+\frac{(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}","-\frac{2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac{a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (8 A+7 B)+\frac{(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}",1,"(7*a^4*(8*A + 7*B)*x)/16 + (4*a^4*(8*A + 7*B)*Sin[c + d*x])/(5*d) + (27*a^4*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(8*A + 7*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*A - B)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (B*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(8*A + 7*B)*Sin[c + d*x]^3)/(15*d)","A",13,8,29,0.2759,1,"{2968, 3023, 2751, 2645, 2637, 2635, 8, 2633}"
30,1,150,0,0.1390835,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","-\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (5 A+4 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}","-\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (5 A+4 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(7*a^4*(5*A + 4*B)*x)/8 + (8*a^4*(5*A + 4*B)*Sin[c + d*x])/(5*d) + (27*a^4*(5*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^4*(5*A + 4*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) - (4*a^4*(5*A + 4*B)*Sin[c + d*x]^3)/(15*d)","A",11,6,23,0.2609,1,"{2751, 2645, 2637, 2635, 8, 2633}"
31,1,151,0,0.4126117,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{(4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{(32 A+35 B) \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)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{(4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{(32 A+35 B) \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)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^4*(48*A + 35*B)*x)/8 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(8*A + 7*B)*Sin[c + d*x])/(8*d) + (a*B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + ((32*A + 35*B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d)","A",7,5,29,0.1724,1,"{2976, 2968, 3023, 2735, 3770}"
32,1,150,0,0.4532214,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(3 A-B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}-\frac{(3 A-8 B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (13 A+12 B)+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}","\frac{5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(3 A-B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}-\frac{(3 A-8 B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (13 A+12 B)+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^4*(13*A + 12*B)*x)/2 + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A + 2*B)*Sin[c + d*x])/(2*d) - ((3*A - B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) - ((3*A - 8*B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d","A",7,6,31,0.1935,1,"{2975, 2976, 2968, 3023, 2735, 3770}"
33,1,162,0,0.475574,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","-\frac{5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A+B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{2 d}+\frac{(5 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 d}+\frac{1}{2} a^4 x (8 A+13 B)+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}","-\frac{5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A+B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{2 d}+\frac{(5 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 d}+\frac{1}{2} a^4 x (8 A+13 B)+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^4*(8*A + 13*B)*x)/2 + (a^4*(13*A + 8*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B)*Sin[c + d*x])/(2*d) - ((6*A + B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(2*d) + ((5*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,31,0.1935,1,"{2975, 2976, 2968, 3023, 2735, 3770}"
34,1,165,0,0.5141027,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","-\frac{5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(11 A+9 B) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+\frac{(2 A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 d}+a^4 x (A+4 B)+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}","-\frac{5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(11 A+9 B) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+\frac{(2 A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 d}+a^4 x (A+4 B)+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"a^4*(A + 4*B)*x + (a^4*(12*A + 13*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B)*Sin[c + d*x])/(2*d) + ((11*A + 9*B)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(3*d) + ((2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,5,31,0.1613,1,"{2975, 2968, 3023, 2735, 3770}"
35,1,173,0,0.5228709,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(7 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{(35 A+32 B) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 B x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}","\frac{5 a^4 (7 A+8 B) \tan (c+d x)}{8 d}+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(7 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{(35 A+32 B) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 B x+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"a^4*B*x + (a^4*(35*A + 48*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*(7*A + 8*B)*Tan[c + d*x])/(8*d) + ((35*A + 32*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((7*A + 4*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,5,31,0.1613,1,"{2975, 2968, 3021, 2735, 3770}"
36,1,198,0,0.587073,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(8 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 d}+\frac{(26 A+25 B) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{30 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}","\frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(8 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 d}+\frac{(26 A+25 B) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{30 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(7*a^4*(4*A + 5*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(83*A + 100*B)*Tan[c + d*x])/(15*d) + (a^4*(244*A + 275*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((26*A + 25*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((8*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",9,7,31,0.2258,1,"{2975, 2968, 3021, 2748, 3767, 8, 3770}"
37,1,229,0,0.6496477,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]","\frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 d}+\frac{(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}","\frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 d}+\frac{(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"(7*a^4*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B)*Tan[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((73*A + 72*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",10,8,31,0.2581,1,"{2975, 2968, 3021, 2748, 3768, 3770, 3767, 8}"
38,1,153,0,0.206986,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","-\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}+\frac{4 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(4 A-5 B) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 (4 A-5 B) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x (4 A-5 B)}{8 a}","-\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}+\frac{4 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(4 A-5 B) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 (4 A-5 B) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x (4 A-5 B)}{8 a}",1,"(-3*(4*A - 5*B)*x)/(8*a) + (4*(A - B)*Sin[c + d*x])/(a*d) - (3*(4*A - 5*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - ((4*A - 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (4*(A - B)*Sin[c + d*x]^3)/(3*a*d)","A",7,5,31,0.1613,1,"{2977, 2748, 2633, 2635, 8}"
39,1,122,0,0.1715394,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","\frac{(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (A-B)}{2 a}","\frac{(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (A-B)}{2 a}",1,"(3*(A - B)*x)/(2*a) - ((3*A - 4*B)*Sin[c + d*x])/(a*d) + (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,31,0.1613,1,"{2977, 2748, 2635, 8, 2633}"
40,1,99,0,0.1229903,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","\frac{2 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B)}{2 a}","\frac{(A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x (A-B)}{a}+\frac{B \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{B x}{2 a}",1,"-((2*A - 3*B)*x)/(2*a) + (2*(A - B)*Sin[c + d*x])/(a*d) - ((2*A - 3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",2,2,31,0.06452,1,"{2977, 2734}"
41,1,54,0,0.1389214,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","-\frac{(A-B) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (A-B)}{a}+\frac{B \sin (c+d x)}{a d}","-\frac{(A-B) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (A-B)}{a}+\frac{B \sin (c+d x)}{a d}",1,"((A - B)*x)/a + (B*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))","A",5,5,29,0.1724,1,"{2968, 3023, 12, 2735, 2648}"
42,1,34,0,0.0498775,"\int \frac{A+B \cos (c+d x)}{a+a \cos (c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{B x}{a}","\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{B x}{a}",1,"(B*x)/a + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",2,2,23,0.08696,1,"{2735, 2648}"
43,1,44,0,0.0783777,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",3,3,29,0.1034,1,"{2978, 12, 3770}"
44,1,69,0,0.1562321,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{(2 A-B) \tan (c+d x)}{a d}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \tan (c+d x)}{d (a \cos (c+d x)+a)}","\frac{(2 A-B) \tan (c+d x)}{a d}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"-(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B)*Tan[c + d*x])/(a*d) - ((A - B)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{2978, 2748, 3767, 8, 3770}"
45,1,107,0,0.1691357,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","-\frac{2 (A-B) \tan (c+d x)}{a d}+\frac{(3 A-2 B) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}","-\frac{2 (A-B) \tan (c+d x)}{a d}+\frac{(3 A-2 B) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"((3*A - 2*B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(A - B)*Tan[c + d*x])/(a*d) + ((3*A - 2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{2978, 2748, 3768, 3770, 3767, 8}"
46,1,131,0,0.1795602,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{(4 A-3 B) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B) \tan (c+d x)}{a d}-\frac{3 (A-B) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (A-B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}","\frac{(4 A-3 B) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B) \tan (c+d x)}{a d}-\frac{3 (A-B) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (A-B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(-3*(A - B)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((4*A - 3*B)*Tan[c + d*x])/(a*d) - (3*(A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A - 3*B)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,31,0.1613,1,"{2978, 2748, 3767, 3768, 3770}"
47,1,170,0,0.322422,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{4 (2 A-3 B) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 A-3 B) \sin (c+d x)}{a^2 d}+\frac{(7 A-10 B) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 A-10 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 A-10 B)}{2 a^2}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{4 (2 A-3 B) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 A-3 B) \sin (c+d x)}{a^2 d}+\frac{(7 A-10 B) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 A-10 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 A-10 B)}{2 a^2}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A - 10*B)*x)/(2*a^2) - (4*(2*A - 3*B)*Sin[c + d*x])/(a^2*d) + ((7*A - 10*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((7*A - 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(2*A - 3*B)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,5,31,0.1613,1,"{2977, 2748, 2635, 8, 2633}"
48,1,147,0,0.3412508,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{2 (5 A-8 B) \sin (c+d x)}{3 a^2 d}+\frac{(5 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B)}{2 a^2}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 A-8 B) \sin (c+d x)}{3 a^2 d}+\frac{(5 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B)}{2 a^2}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-((4*A - 7*B)*x)/(2*a^2) + (2*(5*A - 8*B)*Sin[c + d*x])/(3*a^2*d) - ((4*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((5*A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,2,31,0.06452,1,"{2977, 2734}"
49,1,99,0,0.2757463,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}-\frac{(A-2 B) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (A-2 B)}{a^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}-\frac{(A-2 B) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (A-2 B)}{a^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((A - 2*B)*x)/a^2 - ((A - 4*B)*Sin[c + d*x])/(3*a^2*d) - ((A - 2*B)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*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,"{2977, 2968, 3023, 12, 2735, 2648}"
50,1,70,0,0.1551692,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{(2 A-5 B) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B x}{a^2}-\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 A-5 B) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B x}{a^2}-\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(B*x)/a^2 + ((2*A - 5*B)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",4,4,29,0.1379,1,"{2968, 3019, 2735, 2648}"
51,1,65,0,0.0537473,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2,x]","\frac{(A+2 B) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(A+2 B) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A + 2*B)*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))","A",2,2,23,0.08696,1,"{2750, 2648}"
52,1,79,0,0.1796284,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","-\frac{(4 A-B) \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) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{(4 A-B) \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) \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)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",4,3,29,0.1034,1,"{2978, 12, 3770}"
53,1,107,0,0.2952814,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{2 (5 A-2 B) \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) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 A-2 B) \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) \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)) + (2*(5*A - 2*B)*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)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,5,31,0.1613,1,"{2978, 2748, 3767, 8, 3770}"
54,1,152,0,0.3136634,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (8 A-5 B) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{2 (8 A-5 B) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((7*A - 4*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*A - 5*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,31,0.1935,1,"{2978, 2748, 3768, 3770, 3767, 8}"
55,1,179,0,0.3649721,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{4 (3 A-2 B) \tan ^3(c+d x)}{3 a^2 d}+\frac{4 (3 A-2 B) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{4 (3 A-2 B) \tan ^3(c+d x)}{3 a^2 d}+\frac{4 (3 A-2 B) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-((10*A - 7*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + (4*(3*A - 2*B)*Tan[c + d*x])/(a^2*d) - ((10*A - 7*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((10*A - 7*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(3*A - 2*B)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,5,31,0.1613,1,"{2978, 2748, 3767, 3768, 3770}"
56,1,218,0,0.5154348,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d}-\frac{4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac{(13 A-23 B) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(13 A-23 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (13 A-23 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(8 A-13 B) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d}-\frac{4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac{(13 A-23 B) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(13 A-23 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (13 A-23 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(8 A-13 B) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((13*A - 23*B)*x)/(2*a^3) - (4*(19*A - 34*B)*Sin[c + d*x])/(5*a^3*d) + ((13*A - 23*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((8*A - 13*B)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((13*A - 23*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(19*A - 34*B)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,5,31,0.1613,1,"{2977, 2748, 2635, 8, 2633}"
57,1,193,0,0.4676782,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{8 (9 A-19 B) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{8 (9 A-19 B) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((6*A - 13*B)*x)/(2*a^3) + (8*(9*A - 19*B)*Sin[c + d*x])/(15*a^3*d) - ((6*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*A - 19*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,2,31,0.06452,1,"{2977, 2734}"
58,1,147,0,0.4570308,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","-\frac{(7 A-27 B) \sin (c+d x)}{15 a^3 d}-\frac{(A-3 B) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (A-3 B)}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{(7 A-27 B) \sin (c+d x)}{15 a^3 d}-\frac{(A-3 B) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (A-3 B)}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((A - 3*B)*x)/a^3 - ((7*A - 27*B)*Sin[c + d*x])/(15*a^3*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 3*B)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,31,0.1935,1,"{2977, 2968, 3023, 12, 2735, 2648}"
59,1,116,0,0.3210751,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(4 A-29 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B x}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(4 A-29 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B x}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(B*x)/a^3 + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - 7*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((4*A - 29*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{2977, 2968, 3019, 2735, 2648}"
60,1,102,0,0.1880356,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(3 A+7 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-8 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(3 A+7 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-8 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A + 7*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,4,29,0.1379,1,"{2968, 3019, 2750, 2648}"
61,1,102,0,0.0788386,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3,x]","\frac{(2 A+3 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{(2 A+3 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,23,0.1304,1,"{2750, 2650, 2648}"
62,1,117,0,0.3109936,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{2 (11 A-B) \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) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{2 (11 A-B) \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) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \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)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*A - B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,3,29,0.1034,1,"{2978, 12, 3770}"
63,1,145,0,0.4693333,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{2 (36 A-11 B) \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) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{2 (36 A-11 B) \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) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \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)*Tan[c + d*x])/(15*a^3*d) - ((A - B)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B)*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",7,5,31,0.1613,1,"{2978, 2748, 3767, 8, 3770}"
64,1,196,0,0.541149,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","-\frac{8 (19 A-9 B) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \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) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{8 (19 A-9 B) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \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) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((13*A - 6*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*A - 9*B)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*A - 9*B)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,31,0.1935,1,"{2978, 2748, 3768, 3770, 3767, 8}"
65,1,229,0,0.6720425,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^5*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]","\frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac{(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{x (8 A-21 B)}{2 a^4}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}","\frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac{(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{x (8 A-21 B)}{2 a^4}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"-((8*A - 21*B)*x)/(2*a^4) + (8*(83*A - 216*B)*Sin[c + d*x])/(105*a^4*d) - ((8*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) + ((52*A - 129*B)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*(83*A - 216*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((A - 2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",5,2,31,0.06452,1,"{2977, 2734}"
66,1,185,0,0.6791832,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]","-\frac{(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac{(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (A-4 B)}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(5 A-12 B) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","-\frac{(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac{(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (A-4 B)}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(5 A-12 B) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"((A - 4*B)*x)/a^4 - ((55*A - 244*B)*Sin[c + d*x])/(105*a^4*d) + ((25*A - 88*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((A - 4*B)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A - 12*B)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,31,0.1935,1,"{2977, 2968, 3023, 12, 2735, 2648}"
67,1,154,0,0.4981074,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]","\frac{(12 A-215 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(6 A-55 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{B x}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(3 A-10 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(12 A-215 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(6 A-55 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{B x}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(3 A-10 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(B*x)/a^4 - ((6*A - 55*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((12*A - 215*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 10*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,5,31,0.1613,1,"{2977, 2968, 3019, 2735, 2648}"
68,1,136,0,0.3476085,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]","\frac{(13 A+36 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{2 (A+27 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(A-8 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{(13 A+36 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{2 (A+27 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(A-8 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(-2*(A + 27*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((13*A + 36*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((A - 8*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",5,5,31,0.1613,1,"{2977, 2968, 3019, 2750, 2648}"
69,1,138,0,0.2146903,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]","\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"-((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((4*A - 11*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((8*A + 13*B)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((8*A + 13*B)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",5,5,29,0.1724,1,"{2968, 3019, 2750, 2650, 2648}"
70,1,138,0,0.1379884,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4,x]","\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",4,3,23,0.1304,1,"{2750, 2650, 2648}"
71,1,147,0,0.4656866,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^4,x]","-\frac{2 (80 A-3 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B) \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) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","-\frac{2 (80 A-3 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B) \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) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \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)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (2*(80*A - 3*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",6,3,29,0.1034,1,"{2978, 12, 3770}"
72,1,175,0,0.6716463,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\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{(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\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{(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \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)) + (8*(83*A - 20*B)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 25*B)*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)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,5,31,0.1613,1,"{2978, 2748, 3767, 8, 3770}"
73,1,232,0,0.6876156,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}","-\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((21*A - 8*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A - 52*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(216*A - 83*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B)*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",9,6,31,0.1935,1,"{2978, 2748, 3768, 3770, 3767, 8}"
74,1,187,0,0.3039986,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 a (9 A+8 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (9 A+8 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{4 a (9 A+8 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (9 A+8 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (9 A+8 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{4 a (9 A+8 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*(9*A + 8*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*A + 8*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(9*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (4*(9*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)","A",5,5,33,0.1515,1,"{2981, 2770, 2759, 2751, 2646}"
75,1,144,0,0.2648052,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 (7 A+6 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (7 A+6 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*(7*A + 6*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(7*A + 6*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)","A",4,4,33,0.1212,1,"{2981, 2759, 2751, 2646}"
76,1,101,0,0.2018728,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 (5 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}","\frac{2 (5 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(2*a*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",4,4,31,0.1290,1,"{2968, 3023, 2751, 2646}"
77,1,62,0,0.0586005,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 a (3 A+B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{2 a (3 A+B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(2*a*(3*A + B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",2,2,25,0.08000,1,"{2751, 2646}"
78,1,66,0,0.1375953,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 B \sin (c+d x)}{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 a B \sin (c+d x)}{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*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,31,0.09677,1,"{2981, 2773, 206}"
79,1,68,0,0.162592,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","\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 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,33,0.09091,1,"{2980, 2773, 206}"
80,1,117,0,0.2203563,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a (3 A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (3 A+4 B) \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) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{a (3 A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (3 A+4 B) \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) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(3*A + 4*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(3*A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,33,0.1212,1,"{2980, 2772, 2773, 206}"
81,1,160,0,0.2927186,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a (5 A+6 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B) \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)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{a (5 A+6 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B) \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)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(5*A + 6*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 6*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",5,4,33,0.1212,1,"{2980, 2772, 2773, 206}"
82,1,234,0,0.5293236,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 A+168 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}","\frac{2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 A+168 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(4*a^2*(187*A + 168*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(187*A + 168*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(11*A + 12*B)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*a*(187*A + 168*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*B*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(187*A + 168*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d)","A",6,6,33,0.1818,1,"{2976, 2981, 2770, 2759, 2751, 2646}"
83,1,189,0,0.446453,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 a^2 (9 A+10 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (39 A+34 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (39 A+34 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{4 a (39 A+34 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}","\frac{2 a^2 (9 A+10 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (39 A+34 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (39 A+34 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{4 a (39 A+34 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(2*a^2*(39*A + 34*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(9*A + 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(39*A + 34*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*B*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(39*A + 34*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)","A",5,5,33,0.1515,1,"{2976, 2981, 2759, 2751, 2646}"
84,1,138,0,0.2503522,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{8 a^2 (21 A+19 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 A+19 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}","\frac{8 a^2 (21 A+19 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 A+19 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}",1,"(8*a^2*(21*A + 19*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*A + 19*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A - 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",5,5,31,0.1613,1,"{2968, 3023, 2751, 2647, 2646}"
85,1,101,0,0.0866019,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(8*a^2*(5*A + 3*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",3,3,25,0.1200,1,"{2751, 2647, 2646}"
86,1,105,0,0.2652831,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 a^2 (3 A+4 B) \sin (c+d x)}{3 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 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{2 a^2 (3 A+4 B) \sin (c+d x)}{3 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 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 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*(3*A + 4*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,31,0.1290,1,"{2976, 2981, 2773, 206}"
87,1,103,0,0.2840825,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{a^2 (A-2 B) \sin (c+d x)}{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 A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}","-\frac{a^2 (A-2 B) \sin (c+d x)}{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 A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{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*(A - 2*B)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d","A",4,4,33,0.1212,1,"{2975, 2981, 2773, 206}"
88,1,119,0,0.3252174,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a^2 (5 A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B) \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) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{a^2 (5 A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (7 A+12 B) \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) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(a^(3/2)*(7*A + 12*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B)*Tan[c + d*x])/(4*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])/(2*d)","A",4,4,33,0.1212,1,"{2975, 2980, 2773, 206}"
89,1,164,0,0.4002163,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a^2 (11 A+14 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+14 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (7 A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{a^2 (11 A+14 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (11 A+14 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (7 A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(a^(3/2)*(11*A + 14*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(11*A + 14*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*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])/(3*d)","A",5,5,33,0.1515,1,"{2975, 2980, 2772, 2773, 206}"
90,1,209,0,0.4846836,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+88 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (9 A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (75 A+88 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}","\frac{a^2 (75 A+88 B) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (75 A+88 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (9 A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (75 A+88 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(a^(3/2)*(75*A + 88*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(9*A + 8*B)*Sec[c + d*x]^2*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]^3*Tan[c + d*x])/(4*d)","A",6,5,33,0.1515,1,"{2975, 2980, 2772, 2773, 206}"
91,1,237,0,0.6480122,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{2 a^3 (209 A+194 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (11 A+14 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 a^3 (803 A+710 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (803 A+710 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (803 A+710 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}","\frac{2 a^3 (209 A+194 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (11 A+14 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 a^3 (803 A+710 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}-\frac{4 a^2 (803 A+710 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (803 A+710 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(2*a^3*(803*A + 710*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(209*A + 194*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(803*A + 710*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a^2*(11*A + 14*B)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*(803*A + 710*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",6,5,33,0.1515,1,"{2976, 2981, 2759, 2751, 2646}"
92,1,175,0,0.2796686,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{16 a^2 (15 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}","\frac{16 a^2 (15 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (9 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}",1,"(64*a^3*(15*A + 13*B)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*A - 2*B)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*B*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)","A",6,5,31,0.1613,1,"{2968, 3023, 2751, 2647, 2646}"
93,1,138,0,0.1089798,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(64*a^3*(7*A + 5*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*A + 5*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",4,3,25,0.1200,1,"{2751, 2647, 2646}"
94,1,142,0,0.4140529,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 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 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{2 a^3 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 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 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 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*(35*A + 32*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(5*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",5,4,31,0.1290,1,"{2976, 2981, 2773, 206}"
95,1,144,0,0.4471458,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a^3 (3 A+14 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (3 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 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 A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}","\frac{a^3 (3 A+14 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (3 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 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 A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{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*(3*A + 14*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(3*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",5,5,33,0.1515,1,"{2975, 2976, 2981, 2773, 206}"
96,1,156,0,0.4738597,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","-\frac{a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (7 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a^{5/2} (19 A+20 B) \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) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}","-\frac{a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (7 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a^{5/2} (19 A+20 B) \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) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a^(5/2)*(19*A + 20*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(9*A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,33,0.1212,1,"{2975, 2981, 2773, 206}"
97,1,164,0,0.52581,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a^3 (49 A+54 B) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (25 A+38 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (3 A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^3 (49 A+54 B) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (25 A+38 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (3 A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(25*A + 38*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(3*A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,4,33,0.1212,1,"{2975, 2980, 2773, 206}"
98,1,209,0,0.6086206,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a^3 (163 A+200 B) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+200 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (11 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^3 (95 A+104 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}","\frac{a^3 (163 A+200 B) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (163 A+200 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (11 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^3 (95 A+104 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^(5/2)*(163*A + 200*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(163*A + 200*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(95*A + 104*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(11*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,33,0.1515,1,"{2975, 2980, 2772, 2773, 206}"
99,1,254,0,0.7130666,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a^3 (283 A+326 B) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+326 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (13 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a^3 (157 A+170 B) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (283 A+326 B) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{a^3 (283 A+326 B) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{5/2} (283 A+326 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (13 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a^3 (157 A+170 B) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (283 A+326 B) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^(5/2)*(283*A + 326*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 326*B)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(157*A + 170*B)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",7,5,33,0.1515,1,"{2975, 2980, 2772, 2773, 206}"
100,1,202,0,0.5776391,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (7 A-B) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (7 A-31 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}+\frac{4 (49 A-37 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (7 A-B) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (7 A-31 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}+\frac{4 (49 A-37 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*A - 37*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(7*A - 31*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)","A",7,6,33,0.1818,1,"{2983, 2968, 3023, 2751, 2649, 206}"
101,1,159,0,0.3844849,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (5 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}-\frac{4 (5 A-7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 (5 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}-\frac{4 (5 A-7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(5*A - 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)","A",6,6,33,0.1818,1,"{2983, 2968, 3023, 2751, 2649, 206}"
102,1,118,0,0.2095527,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 (3 A-2 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}","\frac{2 (3 A-2 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"-((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*A - 2*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",5,5,31,0.1613,1,"{2968, 3023, 2751, 2649, 206}"
103,1,78,0,0.0712777,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (A-B) \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 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{2} (A-B) \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 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2751, 2649, 206}"
104,1,91,0,0.1656825,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\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{\sqrt{2} (A-B) \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{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)","A",5,4,31,0.1290,1,"{2985, 2649, 206, 2773}"
105,1,119,0,0.3083128,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","-\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{\sqrt{2} (A-B) \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-2 B) \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-B) \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}}",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)*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",6,5,33,0.1515,1,"{2984, 2985, 2649, 206, 2773}"
106,1,165,0,0.4803717,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{(7 A-4 B) \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) \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 (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","-\frac{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{(7 A-4 B) \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) \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 (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((7*A - 4*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B)*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",7,5,33,0.1515,1,"{2984, 2985, 2649, 206, 2773}"
107,1,261,0,0.7857248,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(273 A-397 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(15 A-19 B) \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) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(7 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}+\frac{(63 A-67 B) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(273 A-397 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(15 A-19 B) \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) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(7 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}+\frac{(63 A-67 B) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"-((15*A - 19*B)*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)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((651*A - 799*B)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((63*A - 67*B)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((273*A - 397*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)","A",8,7,33,0.2121,1,"{2977, 2983, 2968, 3023, 2751, 2649, 206}"
108,1,216,0,0.5949895,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(35 A-39 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}+\frac{(11 A-15 B) \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) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(5 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(65 A-93 B) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}","\frac{(35 A-39 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}+\frac{(11 A-15 B) \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) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(5 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(65 A-93 B) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 15*B)*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)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((65*A - 93*B)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((35*A - 39*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)","A",7,7,33,0.2121,1,"{2977, 2983, 2968, 3023, 2751, 2649, 206}"
109,1,171,0,0.4204562,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(3 A-7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}-\frac{(7 A-11 B) \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) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(9 A-13 B) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(3 A-7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}-\frac{(7 A-11 B) \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) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(9 A-13 B) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"-((7*A - 11*B)*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)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 13*B)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((3*A - 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)","A",6,6,33,0.1818,1,"{2977, 2968, 3023, 2751, 2649, 206}"
110,1,118,0,0.2231797,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 A-7 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 B \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}","\frac{(3 A-7 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 B \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((3*A - 7*B)*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)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,31,0.1613,1,"{2968, 3019, 2751, 2649, 206}"
111,1,87,0,0.0768915,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A+3 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(A+3 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((A + 3*B)*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)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",3,3,25,0.1200,1,"{2750, 2649, 206}"
112,1,127,0,0.3154649,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(5 A-B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(5 A-B) \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) \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)*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)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,31,0.1613,1,"{2978, 2985, 2649, 206, 2773}"
113,1,170,0,0.5209336,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","-\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{(9 A-5 B) \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-B) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\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{(9 A-5 B) \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-B) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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)*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)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,33,0.1818,1,"{2978, 2984, 2985, 2649, 206, 2773}"
114,1,221,0,0.7052559,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A-12 B) \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) \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) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(19 A-12 B) \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) \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) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A - 12*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B)*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)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,33,0.1818,1,"{2978, 2984, 2985, 2649, 206, 2773}"
115,1,261,0,0.7988461,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(85 A-157 B) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(475 A-787 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-283 B) \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) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(13 A-21 B) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","-\frac{(85 A-157 B) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(475 A-787 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(163 A-283 B) \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) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(13 A-21 B) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((163*A - 283*B)*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)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((13*A - 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((985*A - 1729*B)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((85*A - 157*B)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((475*A - 787*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)","A",8,7,33,0.2121,1,"{2977, 2983, 2968, 3023, 2751, 2649, 206}"
116,1,216,0,0.611347,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(39 A-95 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}+\frac{(93 A-197 B) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-163 B) \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) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(9 A-17 B) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","-\frac{(39 A-95 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}+\frac{(93 A-197 B) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-163 B) \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) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(9 A-17 B) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"-((75*A - 163*B)*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)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((9*A - 17*B)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((93*A - 197*B)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 95*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)","A",7,6,33,0.1818,1,"{2977, 2968, 3023, 2751, 2649, 206}"
117,1,169,0,0.420275,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(A-9 B) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B) \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) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","-\frac{(A-9 B) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B) \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) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((19*A - 75*B)*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)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 13*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 9*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2977, 2968, 3019, 2751, 2649, 206}"
118,1,126,0,0.2297027,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(5 A+19 B) \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-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(5 A+19 B) \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-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((5*A + 19*B)*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)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 13*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,31,0.1613,1,"{2968, 3019, 2750, 2649, 206}"
119,1,126,0,0.1028603,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A+5 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(3 A+5 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((3*A + 5*B)*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)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{2750, 2650, 2649, 206}"
120,1,164,0,0.4655478,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{(43 A-3 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{(43 A-3 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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)*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)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,5,31,0.1613,1,"{2978, 2985, 2649, 206, 2773}"
121,1,207,0,0.7147809,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(35 A-11 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\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{(115 A-43 B) \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{(15 A-7 B) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(35 A-11 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\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{(115 A-43 B) \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{(15 A-7 B) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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)*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)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,33,0.1818,1,"{2978, 2984, 2985, 2649, 206, 2773}"
122,1,264,0,0.9229918,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{7 (9 A-5 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A-20 B) \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) \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) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","-\frac{7 (9 A-5 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(39 A-20 B) \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) \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) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((39*A - 20*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (7*(9*A - 5*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,33,0.1818,1,"{2978, 2984, 2985, 2649, 206, 2773}"
123,1,159,0,0.1990404,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{10 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a (9 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{10 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a (9 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*a*(9*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(A + B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,6,31,0.1935,1,"{2968, 3023, 2748, 2635, 2639, 2641}"
124,1,132,0,0.1753106,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{2 a (7 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 a (7 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(6*a*(A + B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,6,31,0.1935,1,"{2968, 3023, 2748, 2635, 2641, 2639}"
125,1,101,0,0.1582975,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*a*(5*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,31,0.1935,1,"{2968, 3023, 2748, 2639, 2635, 2641}"
126,1,70,0,0.1444551,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 a (3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a (3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{2968, 3023, 2748, 2641, 2639}"
127,1,66,0,0.1479271,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,31,0.1613,1,"{2968, 3021, 2748, 2641, 2639}"
128,1,95,0,0.169493,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 a (A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+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) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*B)*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",6,6,31,0.1935,1,"{2968, 3021, 2748, 2636, 2639, 2641}"
129,1,132,0,0.1770307,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 B) \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+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 B) \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)}",1,"(-2*a*(3*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*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)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,31,0.1935,1,"{2968, 3021, 2748, 2636, 2641, 2639}"
130,1,194,0,0.3188225,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{4 a^2 (6 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (9 A+8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (9 A+11 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{4 a^2 (9 A+8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{9 d}","\frac{4 a^2 (6 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (9 A+8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (9 A+11 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{4 a^2 (9 A+8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{9 d}",1,"(4*a^2*(9*A + 8*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(6*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^2*(9*A + 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(9*A + 11*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*B*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(9*d)","A",8,7,33,0.2121,1,"{2976, 2968, 3023, 2748, 2635, 2641, 2639}"
131,1,161,0,0.2887488,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{4 a^2 (7 A+6 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (4 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d}","\frac{4 a^2 (7 A+6 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (4 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d}",1,"(4*a^2*(4*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*B*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",7,7,33,0.2121,1,"{2976, 2968, 3023, 2748, 2639, 2635, 2641}"
132,1,126,0,0.2742847,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{4 a^2 (2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (5 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 B \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 (2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (5 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(4*a^2*(5*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",6,6,33,0.1818,1,"{2976, 2968, 3023, 2748, 2641, 2639}"
133,1,118,0,0.2687398,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{4 a^2 (3 A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d \sqrt{\cos (c+d x)}}+\frac{4 a^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{4 a^2 (3 A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d \sqrt{\cos (c+d x)}}+\frac{4 a^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(4*a^2*B*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(3*A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(3*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2975, 2968, 3023, 2748, 2641, 2639}"
134,1,120,0,0.2801933,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{4 a^2 (2 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (5 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{4 a^2 (2 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (5 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^2*A*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(5*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,33,0.1818,1,"{2975, 2968, 3021, 2748, 2641, 2639}"
135,1,159,0,0.305887,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{4 a^2 (A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (4 A+5 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^2 (A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (4 A+5 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (4*a^2*(4*A + 5*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,33,0.2121,1,"{2975, 2968, 3021, 2748, 2636, 2639, 2641}"
136,1,194,0,0.3374909,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{4 a^2 (6 A+7 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 (6 A+7 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (9 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{4 a^2 (6 A+7 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 (6 A+7 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (9 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^2*(3*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(9*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (4*a^2*(6*A + 7*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,7,33,0.2121,1,"{2975, 2968, 3021, 2748, 2636, 2641, 2639}"
137,1,237,0,0.4792026,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{4 a^3 (121 A+105 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (17 A+15 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{20 a^3 (22 A+21 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^3 (17 A+15 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 (11 A+15 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{99 d}+\frac{4 a^3 (121 A+105 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}","\frac{4 a^3 (121 A+105 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (17 A+15 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{20 a^3 (22 A+21 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^3 (17 A+15 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 (11 A+15 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{99 d}+\frac{4 a^3 (121 A+105 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"(4*a^3*(17*A + 15*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(121*A + 105*B)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 105*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(17*A + 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (20*a^3*(22*A + 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(11*A + 15*B)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(99*d)","A",9,7,33,0.2121,1,"{2976, 2968, 3023, 2748, 2635, 2641, 2639}"
138,1,204,0,0.447287,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{4 a^3 (13 A+11 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (21 A+17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (9 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}","\frac{4 a^3 (13 A+11 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (21 A+17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (9 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(4*a^3*(21*A + 17*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(13*A + 11*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^3*(24*A + 23*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*A + 13*B)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(63*d)","A",8,7,33,0.2121,1,"{2976, 2968, 3023, 2748, 2639, 2635, 2641}"
139,1,171,0,0.4271021,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{4 a^3 (21 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (42 A+41 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}+\frac{2 a B \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}","\frac{4 a^3 (21 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (42 A+41 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}+\frac{2 a B \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"(4*a^3*(9*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(21*A + 13*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*A + 11*B)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",7,6,33,0.1818,1,"{2976, 2968, 3023, 2748, 2641, 2639}"
140,1,169,0,0.4341651,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{4 a^3 (5 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}","\frac{4 a^3 (5 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"(4*a^3*(5*A + 9*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(5*A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - B)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",7,7,33,0.2121,1,"{2975, 2976, 2968, 3023, 2748, 2641, 2639}"
141,1,161,0,0.4300755,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{20 a^3 (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (7 A+3 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{20 a^3 (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (7 A+3 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*a^3*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (20*a^3*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(4*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(7*A + 3*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",7,6,33,0.1818,1,"{2975, 2968, 3023, 2748, 2641, 2639}"
142,1,171,0,0.4615657,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{4 a^3 (3 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (9 A+5 B) \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 (21 A+20 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{4 a^3 (3 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (9 A+5 B) \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 (21 A+20 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-4*a^3*(9*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^3*(21*A + 20*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(9*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",7,6,33,0.1818,1,"{2975, 2968, 3021, 2748, 2641, 2639}"
143,1,204,0,0.4907514,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{4 a^3 (13 A+21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (41 A+42 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 A+7 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (7 A+9 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{4 a^3 (13 A+21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (41 A+42 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 A+7 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (7 A+9 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-4*a^3*(7*A + 9*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(41*A + 42*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(7*A + 9*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(11*A + 7*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))","A",8,7,33,0.2121,1,"{2975, 2968, 3021, 2748, 2636, 2639, 2641}"
144,1,237,0,0.5211413,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{4 a^3 (11 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (11 A+13 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (23 A+24 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (13 A+9 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{4 a^3 (11 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (11 A+13 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (23 A+24 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (13 A+9 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-4*a^3*(17*A + 21*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(23*A + 24*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^3*(11*A + 13*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(13*A + 9*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))","A",9,7,33,0.2121,1,"{2975, 2968, 3021, 2748, 2636, 2641, 2639}"
145,1,156,0,0.1987449,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","\frac{5 (A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","\frac{5 (A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(-3*(5*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(A - B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((5*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{2977, 2748, 2635, 2641, 2639}"
146,1,123,0,0.1795698,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","-\frac{(3 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(3 A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","-\frac{(3 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(3 A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(3*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((3*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{2977, 2748, 2639, 2635, 2641}"
147,1,85,0,0.1498218,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]),x]","\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"-(((A - 3*B)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,33,0.1212,1,"{2977, 2748, 2641, 2639}"
148,1,83,0,0.1504435,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,33,0.1212,1,"{2978, 2748, 2641, 2639}"
149,1,119,0,0.1738808,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}","-\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"-(((3*A - B)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",5,5,33,0.1515,1,"{2978, 2748, 2636, 2639, 2641}"
150,1,153,0,0.1948487,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{(5 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 (A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","\frac{(5 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 (A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(3*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*(A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{2978, 2748, 2636, 2641, 2639}"
151,1,203,0,0.406793,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{5 (2 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 (5 A-8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}+\frac{(2 A-3 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{7 (5 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}+\frac{5 (2 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{5 (2 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 (5 A-8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}+\frac{(2 A-3 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{7 (5 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}+\frac{5 (2 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-7*(5*A - 8*B)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) + (5*(2*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (5*(2*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*(5*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((2*A - 3*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,5,33,0.1515,1,"{2977, 2748, 2635, 2641, 2639}"
152,1,166,0,0.3892694,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","-\frac{5 (A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 (A-2 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 (A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 (A-2 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((4*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) - (5*(A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((4*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,5,33,0.1515,1,"{2977, 2748, 2639, 2635, 2641}"
153,1,136,0,0.3136696,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{(2 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(2 A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(2 A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-(((A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,4,33,0.1212,1,"{2977, 2748, 2641, 2639}"
154,1,121,0,0.2767704,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2,x]","\frac{(A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{(A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"-((B*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,33,0.1515,1,"{2977, 2978, 2748, 2641, 2639}"
155,1,121,0,0.3423494,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{(2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{(2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"(A*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,4,33,0.1212,1,"{2978, 2748, 2641, 2639}"
156,1,168,0,0.3593093,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{(5 A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(5 A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{(5 A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(5 A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"-(((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) - ((5*A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",6,5,33,0.1515,1,"{2978, 2748, 2636, 2639, 2641}"
157,1,201,0,0.3637325,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{5 (2 A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{5 (2 A-B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{5 (2 A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{5 (2 A-B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((7*A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (5*(2*A - B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (5*(2*A - B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A - 4*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",7,5,33,0.1515,1,"{2978, 2748, 2636, 2641, 2639}"
158,1,254,0,0.5477624,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(11 A-21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{7 (17 A-33 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{3 (11 A-21 B) \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 (17 A-33 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}+\frac{(11 A-21 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(7 A-12 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(11 A-21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{7 (17 A-33 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{3 (11 A-21 B) \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 (17 A-33 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}+\frac{(11 A-21 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(7 A-12 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(-7*(17*A - 33*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A - 21*B)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A - 21*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (7*(17*A - 33*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) + ((A - B)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((7*A - 12*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*(11*A - 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",8,5,33,0.1515,1,"{2977, 2748, 2635, 2641, 2639}"
159,1,219,0,0.5182745,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","-\frac{(13 A-33 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","-\frac{(13 A-33 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(7*(7*A - 17*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 33*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) + (7*(7*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",7,5,33,0.1515,1,"{2977, 2748, 2639, 2635, 2641}"
160,1,188,0,0.4775712,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(3 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(3 A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(3 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","\frac{(3 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(3 A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(3 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-((9*A - 49*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",6,4,33,0.1212,1,"{2977, 2748, 2641, 2639}"
161,1,180,0,0.4701114,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}","\frac{(A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"-((A + 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{2977, 2978, 2748, 2641, 2639}"
162,1,178,0,0.464352,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+4 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+4 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((A - B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,33,0.1515,1,"{2977, 2978, 2748, 2641, 2639}"
163,1,182,0,0.4800361,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{(3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{(3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((9*A + B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,4,33,0.1212,1,"{2978, 2748, 2641, 2639}"
164,1,221,0,0.5188323,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{(13 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B) \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) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{(13 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B) \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) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"-((49*A - 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - 9*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",7,5,33,0.1515,1,"{2978, 2748, 2636, 2639, 2641}"
165,1,254,0,0.6026672,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{(33 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (17 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{7 (17 A-7 B) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 (17 A-7 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\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{(A-B) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{(33 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (17 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{7 (17 A-7 B) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 (17 A-7 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\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{(A-B) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(7*(17*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - (7*(17*A - 7*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*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) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",8,5,33,0.1515,1,"{2978, 2748, 2636, 2641, 2639}"
166,1,221,0,0.349647,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{a (8 A+7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a (8 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} (8 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{5 a (8 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}","\frac{a (8 A+7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a (8 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} (8 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{5 a (8 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(5*Sqrt[a]*(8*A + 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (5*a*(8*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*(8*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*A + 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])","A",6,4,35,0.1143,1,"{2981, 2770, 2774, 216}"
167,1,176,0,0.2975642,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{a (6 A+5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{a (6 A+5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(6*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*A + 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",5,4,35,0.1143,1,"{2981, 2770, 2774, 216}"
168,1,131,0,0.2271655,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{a} (4 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (4 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(4*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{2981, 2770, 2774, 216}"
169,1,78,0,0.168901,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a} (2 A+B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (2 A+B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(2*A + B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,35,0.08571,1,"{2981, 2774, 216}"
170,1,76,0,0.1646479,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} B \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)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} B \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",3,3,35,0.08571,1,"{2980, 2774, 216}"
171,1,85,0,0.1634629,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 a (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 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 a (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 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(2*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",2,2,35,0.05714,1,"{2980, 2771}"
172,1,130,0,0.2184876,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 a (4 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{2 a (4 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",3,3,35,0.08571,1,"{2980, 2772, 2771}"
173,1,175,0,0.2859498,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{8 a (6 A+7 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (6 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{8 a (6 A+7 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (6 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,3,35,0.08571,1,"{2980, 2772, 2771}"
174,1,227,0,0.5041169,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a^2 (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (88 A+75 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}","\frac{a^2 (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (88 A+75 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(a^(3/2)*(88*A + 75*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(88*A + 75*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(88*A + 75*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{2976, 2981, 2770, 2774, 216}"
175,1,180,0,0.4122125,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (14 A+11 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (14 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (14 A+11 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (14 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(a^(3/2)*(14*A + 11*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(14*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(6*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{2976, 2981, 2770, 2774, 216}"
176,1,133,0,0.3298734,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{a^{3/2} (12 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{a^{3/2} (12 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(a^(3/2)*(12*A + 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,35,0.1143,1,"{2976, 2981, 2774, 216}"
177,1,126,0,0.3310701,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{a^{3/2} (2 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (2 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (2 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (2 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(a^(3/2)*(2*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(2*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(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]])","A",4,4,35,0.1143,1,"{2975, 2981, 2774, 216}"
178,1,125,0,0.3154173,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 a^2 (4 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} B \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}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 (4 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} B \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}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*a^(3/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(4*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,35,0.1143,1,"{2975, 2980, 2774, 216}"
179,1,134,0,0.3397313,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x)}{15 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}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x)}{15 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}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*a^2*(6*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(18*A + 25*B)*Sin[c + d*x])/(15*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]^(5/2))","A",3,3,35,0.08571,1,"{2975, 2980, 2771}"
180,1,181,0,0.4310207,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 a^2 (52 A+63 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (52 A+63 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*a^2*(8*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*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])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,35,0.1143,1,"{2975, 2980, 2772, 2771}"
181,1,228,0,0.5051708,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{8 a^2 (34 A+39 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 a^2 (34 A+39 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^2*(10*A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(34*A + 39*B)*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])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,35,0.1143,1,"{2975, 2980, 2772, 2771}"
182,1,274,0,0.7088424,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a^{5/2} (326 A+283 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a^{5/2} (326 A+283 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^(5/2)*(326*A + 283*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(326*A + 283*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(326*A + 283*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(170*A + 157*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(10*A + 13*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,5,35,0.1429,1,"{2976, 2981, 2770, 2774, 216}"
183,1,227,0,0.7132397,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{a^3 (104 A+95 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^{5/2} (200 A+163 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (200 A+163 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}","\frac{a^3 (104 A+95 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a^{5/2} (200 A+163 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (200 A+163 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^(5/2)*(200*A + 163*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(200*A + 163*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(104*A + 95*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 11*B)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{2976, 2981, 2770, 2774, 216}"
184,1,180,0,0.5470039,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{a^{5/2} (38 A+25 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}","\frac{a^{5/2} (38 A+25 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(38*A + 25*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^3*(54*A + 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,4,35,0.1143,1,"{2976, 2981, 2774, 216}"
185,1,178,0,0.5521386,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{a^{5/2} (20 A+19 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (20 A+19 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^(5/2)*(20*A + 19*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(4*A - 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(4*A - B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2975, 2976, 2981, 2774, 216}"
186,1,173,0,0.5328469,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{a^{5/2} (2 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\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{a^{5/2} (2 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\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)}",1,"(a^(5/2)*(2*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(14*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(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))","A",5,4,35,0.1143,1,"{2975, 2981, 2774, 216}"
187,1,172,0,0.5060728,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 a^2 (8 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^3 (32 A+35 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} B \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}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (8 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^3 (32 A+35 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} B \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}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*a^(5/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(32*A + 35*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*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 + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",5,4,35,0.1143,1,"{2975, 2980, 2774, 216}"
188,1,181,0,0.5518903,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 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^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 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^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3*(10*A + 11*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(230*A + 301*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*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 + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,3,35,0.08571,1,"{2975, 2980, 2771}"
189,1,228,0,0.7034999,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (292 A+345 B) \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) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (292 A+345 B) \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) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*a^3*(124*A + 135*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*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 + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,35,0.1143,1,"{2975, 2980, 2772, 2771}"
190,1,275,0,0.7138626,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(13/2),x]","\frac{8 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (194 A+209 B) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (14 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3 (710 A+803 B) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (194 A+209 B) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (14 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3*(194*A + 209*B)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*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 + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",6,4,35,0.1143,1,"{2975, 2980, 2772, 2771}"
191,1,190,0,0.5947874,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{(4 A-7 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) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","-\frac{(4 A-7 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) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"-((4*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) + (Sqrt[2]*(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)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,35,0.1714,1,"{2983, 2982, 2782, 205, 2774, 216}"
192,1,141,0,0.3965403,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{(2 A-B) \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) \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{(2 A-B) \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) \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"((2*A - B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(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) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{2983, 2982, 2782, 205, 2774, 216}"
193,1,100,0,0.2421463,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} (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{2 B \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) \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 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(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)","A",5,5,35,0.1429,1,"{2982, 2782, 205, 2774, 216}"
194,1,99,0,0.1924375,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"-((Sqrt[2]*(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)) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{2984, 12, 2782, 205}"
195,1,142,0,0.3355266,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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)*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",5,4,35,0.1143,1,"{2984, 12, 2782, 205}"
196,1,187,0,0.6087317,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\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 (13 A-5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","-\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 (13 A-5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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)*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)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,4,35,0.1143,1,"{2984, 12, 2782, 205}"
197,1,197,0,0.6363377,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(2 A-3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}","\frac{(2 A-3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"((2*A - 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - 9*B)*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)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,35,0.2000,1,"{2977, 2983, 2982, 2782, 205, 2774, 216}"
198,1,145,0,0.4031734,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 B \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 B \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B)*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)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,6,35,0.1714,1,"{2977, 2982, 2782, 205, 2774, 216}"
199,1,107,0,0.2163016,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((3*A + B)*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)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,35,0.1143,1,"{2978, 12, 2782, 205}"
200,1,156,0,0.3840136,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - 3*B)*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)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
201,1,203,0,0.5554608,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 7*B)*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)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
202,1,246,0,0.8368389,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(2 A-5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(7 A-15 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(2 A-5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(7 A-15 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((2*A - 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 115*B)*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)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((7*A - 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((11*A - 35*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,35,0.2000,1,"{2977, 2983, 2982, 2782, 205, 2774, 216}"
203,1,194,0,0.5822389,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 B \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) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 B \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) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B)*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)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,35,0.1714,1,"{2977, 2982, 2782, 205, 2774, 216}"
204,1,154,0,0.3769153,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((5*A + 3*B)*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)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,35,0.1429,1,"{2977, 2978, 12, 2782, 205}"
205,1,156,0,0.4390144,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 5*B)*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)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,4,35,0.1143,1,"{2978, 12, 2782, 205}"
206,1,203,0,0.5665601,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"-((75*A - 19*B)*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)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
207,1,250,0,0.7517857,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{(95 A-39 B) \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) \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) \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) \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) \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) \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) \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) \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) \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) \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)*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)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*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)*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)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
208,1,293,0,1.0440206,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{(79 A-259 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(2 A-7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(3 A-7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}","\frac{(79 A-259 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(2 A-7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(3 A-7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}",1,"((2*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((3*A - 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((79*A - 259*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) - (7*(7*A - 27*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,35,0.2000,1,"{2977, 2983, 2982, 2782, 205, 2774, 216}"
209,1,241,0,0.7650321,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{(5 A-49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 B \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(5 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}","\frac{(5 A-49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 B \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(5 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"(2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) + ((5*A - 177*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((5*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",8,6,35,0.1714,1,"{2977, 2982, 2782, 205, 2774, 216}"
210,1,201,0,0.5871764,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{(17 A+67 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}","\frac{(17 A+67 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"((7*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((17*A + 67*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,35,0.1429,1,"{2977, 2978, 12, 2782, 205}"
211,1,201,0,0.5792934,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","-\frac{(5 A-17 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","-\frac{(5 A-17 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"((13*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 17*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,35,0.1429,1,"{2977, 2978, 12, 2782, 205}"
212,1,203,0,0.5914359,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)),x]","-\frac{(103 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","-\frac{(103 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"((63*A + 13*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((5*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((103*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,4,35,0.1143,1,"{2978, 12, 2782, 205}"
213,1,250,0,0.8038487,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}","\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(-3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
214,1,297,0,1.0321108,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"((1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sin[c + d*x])/(192*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((1887*A - 691*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",8,5,35,0.1429,1,"{2978, 2984, 12, 2782, 205}"
215,1,105,0,0.1697509,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(4 a A+3 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a A+3 b B)+\frac{b B \sin (c+d x) \cos ^3(c+d x)}{4 d}","-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(4 a A+3 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a A+3 b B)+\frac{b B \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"((4*a*A + 3*b*B)*x)/8 + ((A*b + a*B)*Sin[c + d*x])/d + ((4*a*A + 3*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)","A",7,6,29,0.2069,1,"{2968, 3023, 2748, 2635, 8, 2633}"
216,1,84,0,0.0896968,"\int \cos (c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{b B \sin (c+d x) \cos ^2(c+d x)}{3 d}","\frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{b B \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"((A*b + a*B)*x)/2 + ((3*a*A + 2*b*B)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",3,3,27,0.1111,1,"{2968, 3023, 2734}"
217,1,52,0,0.0228202,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (2 a A+b B)+\frac{b B \sin (c+d x) \cos (c+d x)}{2 d}","\frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (2 a A+b B)+\frac{b B \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*a*A + b*B)*x)/2 + ((A*b + a*B)*Sin[c + d*x])/d + (b*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,21,0.04762,1,"{2734}"
218,1,35,0,0.1050838,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","x (a B+A b)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \sin (c+d x)}{d}","x (a B+A b)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \sin (c+d x)}{d}",1,"(A*b + a*B)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (b*B*Sin[c + d*x])/d","A",4,4,27,0.1481,1,"{2968, 3023, 2735, 3770}"
219,1,35,0,0.1137085,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*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}+b B x","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+b B x",1,"b*B*x + ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d","A",4,4,29,0.1379,1,"{2968, 3021, 2735, 3770}"
220,1,61,0,0.1478972,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(a A+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(a A+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"((a*A + 2*b*B)*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",6,6,29,0.2069,1,"{2968, 3021, 2748, 3767, 8, 3770}"
221,1,93,0,0.1633211,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{(2 a A+3 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{(2 a A+3 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"((A*b + a*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*A + 3*b*B)*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",7,7,29,0.2414,1,"{2968, 3021, 2748, 3768, 3770, 3767, 8}"
222,1,114,0,0.178587,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{(a B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(3 a A+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a A+4 b B) \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 B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(3 a A+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a A+4 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((3*a*A + 4*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((A*b + a*B)*Tan[c + d*x])/d + ((3*a*A + 4*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A*b + a*B)*Tan[c + d*x]^3)/(3*d)","A",7,6,29,0.2069,1,"{2968, 3021, 2748, 3767, 3768, 3770}"
223,1,189,0,0.3110753,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{\left(4 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 A+6 a b B+3 A b^2\right)-\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin (c+d x)}{5 d}+\frac{b (6 a B+5 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}","\frac{\left(4 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 A+6 a b B+3 A b^2\right)-\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin (c+d x)}{5 d}+\frac{b (6 a B+5 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}",1,"((4*a^2*A + 3*A*b^2 + 6*a*b*B)*x)/8 + ((4*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x])/(5*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*(5*A*b + 6*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) - ((4*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{2990, 3023, 2748, 2635, 8, 2633}"
224,1,170,0,0.2335173,"\int \cos (c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{\left(4 a^2 A b+a^3 (-B)+8 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 b d}+\frac{\left(-2 a^2 B+8 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+3 b^2 B\right)+\frac{(4 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}","\frac{\left(4 a^2 A b+a^3 (-B)+8 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 b d}+\frac{\left(-2 a^2 B+8 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+3 b^2 B\right)+\frac{(4 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"((8*a*A*b + 4*a^2*B + 3*b^2*B)*x)/8 + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*Sin[c + d*x])/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b - a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)","A",4,4,29,0.1379,1,"{2968, 3023, 2753, 2734}"
225,1,107,0,0.0935754,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{2 \left(a^2 B+3 a A b+b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 A+2 a b B+A b^2\right)+\frac{b (2 a B+3 A b) \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 \left(a^2 B+3 a A b+b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 A+2 a b B+A b^2\right)+\frac{b (2 a B+3 A b) \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,"((2*a^2*A + A*b^2 + 2*a*b*B)*x)/2 + (2*(3*a*A*b + a^2*B + b^2*B)*Sin[c + d*x])/(3*d) + (b*(3*A*b + 2*a*B)*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",2,2,23,0.08696,1,"{2753, 2734}"
226,1,86,0,0.176197,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{1}{2} x \left(2 a^2 B+4 a A b+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))}{2 d}","\frac{1}{2} x \left(2 a^2 B+4 a A b+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"((4*a*A*b + 2*a^2*B + b^2*B)*x)/2 + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (b*(2*A*b + 3*a*B)*Sin[c + d*x])/(2*d) + (b*B*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)","A",4,4,29,0.1379,1,"{2990, 3023, 2735, 3770}"
227,1,60,0,0.1685263,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a^2 A \tan (c+d x)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a B+A b)+\frac{b^2 B \sin (c+d x)}{d}","\frac{a^2 A \tan (c+d x)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a B+A b)+\frac{b^2 B \sin (c+d x)}{d}",1,"b*(A*b + 2*a*B)*x + (a*(2*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b^2*B*Sin[c + d*x])/d + (a^2*A*Tan[c + d*x])/d","A",4,4,31,0.1290,1,"{2988, 3023, 2735, 3770}"
228,1,80,0,0.1995098,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\left(a^2 A+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a B+2 A b) \tan (c+d x)}{d}+b^2 B x","\frac{\left(a^2 A+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a B+2 A b) \tan (c+d x)}{d}+b^2 B x",1,"b^2*B*x + ((a^2*A + 2*A*b^2 + 4*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A*b + a*B)*Tan[c + d*x])/d + (a^2*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,31,0.1290,1,"{2988, 3021, 2735, 3770}"
229,1,116,0,0.2701609,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\left(2 a^2 A+6 a b B+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 B+2 a A b+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x)}{2 d}","\frac{\left(2 a^2 A+6 a b B+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 B+2 a A b+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x)}{2 d}",1,"((2*a*A*b + a^2*B + 2*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a^2*A + 3*A*b^2 + 6*a*b*B)*Tan[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{2988, 3021, 2748, 3767, 8, 3770}"
230,1,156,0,0.2934715,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{\left(2 a^2 B+4 a A b+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{\left(2 a^2 B+4 a A b+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"((3*a^2*A + 4*A*b^2 + 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B)*Tan[c + d*x])/(3*d) + ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{2988, 3021, 2748, 3768, 3770, 3767, 8}"
231,1,269,0,0.5074794,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","-\frac{\left(15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{5 d}+\frac{b \left(14 a^2 B+18 a A b+5 b^2 B\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right)+\frac{b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}","-\frac{\left(15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(15 a^2 A b+5 a^3 B+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{5 d}+\frac{b \left(14 a^2 B+18 a A b+5 b^2 B\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right)+\frac{b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}",1,"((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*x)/16 + ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(5*d) + ((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(18*a*A*b + 14*a^2*B + 5*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*(3*A*b + 4*a*B)*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Sin[c + d*x]^3)/(15*d)","A",8,7,31,0.2258,1,"{2990, 3033, 3023, 2748, 2635, 8, 2633}"
232,1,243,0,0.3331539,"\int \cos (c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{\left(15 a^3 A b+52 a^2 b^2 B-3 a^4 B+60 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 b d}+\frac{\left(-3 a^2 B+15 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(30 a^2 A b-6 a^3 B+71 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(12 a^2 A b+4 a^3 B+9 a b^2 B+3 A b^3\right)+\frac{(5 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}","\frac{\left(15 a^3 A b+52 a^2 b^2 B-3 a^4 B+60 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 b d}+\frac{\left(-3 a^2 B+15 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(30 a^2 A b-6 a^3 B+71 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(12 a^2 A b+4 a^3 B+9 a b^2 B+3 A b^3\right)+\frac{(5 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"((12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*x)/8 + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*A*b - a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)","A",5,4,29,0.1379,1,"{2968, 3023, 2753, 2734}"
233,1,171,0,0.1969365,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{\left(16 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{b \left(6 a^2 B+20 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right)+\frac{(3 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{\left(16 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{b \left(6 a^2 B+20 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right)+\frac{(3 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"((8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*x)/8 + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b + 3*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",3,2,23,0.08696,1,"{2753, 2734}"
234,1,137,0,0.3239174,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{b \left(8 a^2 B+9 a A b+2 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(6 a^2 A b+2 a^3 B+3 a b^2 B+A b^3\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 (5 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{b \left(8 a^2 B+9 a A b+2 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(6 a^2 A b+2 a^3 B+3 a b^2 B+A b^3\right)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 (5 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"((6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*x)/2 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*A*b + 8*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*d) + (b^2*(3*A*b + 5*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",5,5,29,0.1724,1,"{2990, 3033, 3023, 2735, 3770}"
235,1,131,0,0.3319308,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{b \left(2 a^2 A-3 a b B-A b^2\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 B+6 a A b+b^2 B\right)+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a A-b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^2}{d}","-\frac{b \left(2 a^2 A-3 a b B-A b^2\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 B+6 a A b+b^2 B\right)+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a A-b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(b*(6*a*A*b + 6*a^2*B + b^2*B)*x)/2 + (a^2*(3*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a^2*A - A*b^2 - 3*a*b*B)*Sin[c + d*x])/d - (b^2*(2*a*A - b*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d","A",5,5,31,0.1613,1,"{2989, 3033, 3023, 2735, 3770}"
236,1,124,0,0.3390243,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a \left(a^2 A+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a B+2 A b) \tan (c+d x)}{d}-\frac{b^2 (a A-2 b B) \sin (c+d x)}{2 d}+b^2 x (3 a B+A b)+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}","\frac{a \left(a^2 A+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a B+2 A b) \tan (c+d x)}{d}-\frac{b^2 (a A-2 b B) \sin (c+d x)}{2 d}+b^2 x (3 a B+A b)+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}",1,"b^2*(A*b + 3*a*B)*x + (a*(a^2*A + 6*A*b^2 + 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a*A - 2*b*B)*Sin[c + d*x])/(2*d) + (a^2*(2*A*b + a*B)*Tan[c + d*x])/d + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,31,0.1613,1,"{2989, 3031, 3023, 2735, 3770}"
237,1,145,0,0.3458143,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 A b+a^3 B+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 B x","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 A b+a^3 B+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 B x",1,"b^3*B*x + ((3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2*A + 8*A*b^2 + 9*a*b*B)*Tan[c + d*x])/(3*d) + (a^2*(5*A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{2989, 3031, 3021, 2735, 3770}"
238,1,188,0,0.4576634,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{\left(6 a^2 A b+2 a^3 B+9 a b^2 B+3 A b^3\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(3 a^2 A+12 a b B+10 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a B+3 A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{\left(6 a^2 A b+2 a^3 B+9 a b^2 B+3 A b^3\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(3 a^2 A+12 a b B+10 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a B+3 A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(3*d) + (a*(3*a^2*A + 10*A*b^2 + 12*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(3*A*b + 2*a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{2989, 3031, 3021, 2748, 3767, 8, 3770}"
239,1,236,0,0.4888125,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{\left(8 a^3 A+30 a^2 b B+30 a A b^2+15 b^3 B\right) \tan (c+d x)}{15 d}+\frac{\left(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(4 a^2 A+15 a b B+12 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (5 a B+7 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\left(8 a^3 A+30 a^2 b B+30 a A b^2+15 b^3 B\right) \tan (c+d x)}{15 d}+\frac{\left(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(4 a^2 A+15 a b B+12 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(9 a^2 A b+3 a^3 B+12 a b^2 B+4 A b^3\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (5 a B+7 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 15*b^3*B)*Tan[c + d*x])/(15*d) + ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(4*a^2*A + 12*A*b^2 + 15*a*b*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(7*A*b + 5*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,31,0.2581,1,"{2989, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
240,1,366,0,0.8376957,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","-\frac{\left(140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right) \sin ^3(c+d x)}{105 d}+\frac{\left(140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right) \sin (c+d x)}{35 d}+\frac{b^2 \left(31 a^2 B+49 a A b+18 b^2 B\right) \sin (c+d x) \cos ^4(c+d x)}{105 d}+\frac{b \left(224 a^2 A b+104 a^3 B+140 a b^2 B+35 A b^3\right) \sin (c+d x) \cos ^3(c+d x)}{168 d}+\frac{\left(36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right)+\frac{b (10 a B+7 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{42 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^3}{7 d}","-\frac{\left(140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right) \sin ^3(c+d x)}{105 d}+\frac{\left(140 a^3 A b+168 a^2 b^2 B+35 a^4 B+112 a A b^3+24 b^4 B\right) \sin (c+d x)}{35 d}+\frac{b^2 \left(31 a^2 B+49 a A b+18 b^2 B\right) \sin (c+d x) \cos ^4(c+d x)}{105 d}+\frac{b \left(224 a^2 A b+104 a^3 B+140 a b^2 B+35 A b^3\right) \sin (c+d x) \cos ^3(c+d x)}{168 d}+\frac{\left(36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(36 a^2 A b^2+8 a^4 A+24 a^3 b B+20 a b^3 B+5 A b^4\right)+\frac{b (10 a B+7 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{42 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^3}{7 d}",1,"((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*x)/16 + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x])/(35*d) + ((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(224*a^2*A*b + 35*A*b^3 + 104*a^3*B + 140*a*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(168*d) + (b^2*(49*a*A*b + 31*a^2*B + 18*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (b*(7*A*b + 10*a*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) - ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x]^3)/(105*d)","A",9,8,31,0.2581,1,"{2990, 3049, 3033, 3023, 2748, 2635, 8, 2633}"
241,1,325,0,0.5092534,"\int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{\left(224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right) \sin (c+d x)}{60 b d}+\frac{\left(-4 a^2 B+24 a A b+25 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{\left(24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}+\frac{\left(48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right)+\frac{(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}","\frac{\left(224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right) \sin (c+d x)}{60 b d}+\frac{\left(-4 a^2 B+24 a A b+25 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{\left(24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}+\frac{\left(48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right)+\frac{(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}",1,"((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x)/16 + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Sin[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (B*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)","A",6,4,29,0.1379,1,"{2968, 3023, 2753, 2734}"
242,1,241,0,0.3377113,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{\left(95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 d}+\frac{\left(12 a^2 B+35 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b \left(130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right)+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}","\frac{\left(95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 d}+\frac{\left(12 a^2 B+35 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b \left(130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right)+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"((8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x)/8 + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*A*b + 4*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",4,2,23,0.08696,1,"{2753, 2734}"
243,1,200,0,0.5470112,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{b \left(34 a^2 A b+19 a^3 B+16 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(26 a^2 B+32 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(32 a^3 A b+24 a^2 b^2 B+8 a^4 B+16 a A b^3+3 b^4 B\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (7 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}","\frac{b \left(34 a^2 A b+19 a^3 B+16 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(26 a^2 B+32 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(32 a^3 A b+24 a^2 b^2 B+8 a^4 B+16 a A b^3+3 b^4 B\right)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (7 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"((32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*x)/8 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (b*(34*a^2*A*b + 4*A*b^3 + 19*a^3*B + 16*a*b^2*B)*Sin[c + d*x])/(6*d) + (b^2*(32*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",6,6,29,0.2069,1,"{2990, 3049, 3033, 3023, 2735, 3770}"
244,1,195,0,0.5696926,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{b \left(6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right) \sin (c+d x)}{3 d}-\frac{b^2 \left(6 a^2 A-8 a b B-3 A b^2\right) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} b x \left(12 a^2 A b+8 a^3 B+4 a b^2 B+A b^3\right)+\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (3 a A-b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^3}{d}","-\frac{b \left(6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right) \sin (c+d x)}{3 d}-\frac{b^2 \left(6 a^2 A-8 a b B-3 A b^2\right) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} b x \left(12 a^2 A b+8 a^3 B+4 a b^2 B+A b^3\right)+\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (3 a A-b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"(b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*x)/2 + (a^3*(4*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(6*a^3*A - 12*a*A*b^2 - 17*a^2*b*B - 2*b^3*B)*Sin[c + d*x])/(3*d) - (b^2*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*a*A - b*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (a*A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d","A",6,6,31,0.1935,1,"{2989, 3049, 3033, 3023, 2735, 3770}"
245,1,209,0,0.6152648,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","-\frac{b \left(13 a^2 A b+4 a^3 B-8 a b^2 B-2 A b^3\right) \sin (c+d x)}{2 d}+\frac{a^2 \left(a^2 A+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \left(2 a^2 B+6 a A b-b^2 B\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2 B+8 a A b+b^2 B\right)+\frac{a (2 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}","-\frac{b \left(13 a^2 A b+4 a^3 B-8 a b^2 B-2 A b^3\right) \sin (c+d x)}{2 d}+\frac{a^2 \left(a^2 A+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \left(2 a^2 B+6 a A b-b^2 B\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2 B+8 a A b+b^2 B\right)+\frac{a (2 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}",1,"(b^2*(8*a*A*b + 12*a^2*B + b^2*B)*x)/2 + (a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(13*a^2*A*b - 2*A*b^3 + 4*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/(2*d) - (b^2*(6*a*A*b + 2*a^2*B - b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,6,31,0.1935,1,"{2989, 3047, 3033, 3023, 2735, 3770}"
246,1,198,0,0.580128,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","-\frac{b^2 \left(3 a^2 B+8 a A b-6 b^2 B\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(2 a^2 A+9 a b B+9 A b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(4 a^2 A b+a^3 B+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+b^3 x (4 a B+A b)+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}","-\frac{b^2 \left(3 a^2 B+8 a A b-6 b^2 B\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(2 a^2 A+9 a b B+9 A b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(4 a^2 A b+a^3 B+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+b^3 x (4 a B+A b)+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"b^3*(A*b + 4*a*B)*x + (a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(8*a*A*b + 3*a^2*B - 6*b^2*B)*Sin[c + d*x])/(6*d) + (a^2*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*Tan[c + d*x])/(3*d) + (a*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,31,0.1935,1,"{2989, 3047, 3031, 3023, 2735, 3770}"
247,1,216,0,0.5974893,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a \left(16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(9 a^2 A+32 a b B+26 A b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x","\frac{a \left(16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(9 a^2 A+32 a b B+26 A b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x",1,"b^4*B*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(16*a^2*A*b + 19*A*b^3 + 4*a^3*B + 34*a*b^2*B)*Tan[c + d*x])/(6*d) + (a^2*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (a*(7*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*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,6,31,0.1935,1,"{2989, 3047, 3031, 3021, 2735, 3770}"
248,1,267,0,0.7229229,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{\left(60 a^2 A b^2+8 a^4 A+40 a^3 b B+60 a b^3 B+15 A b^4\right) \tan (c+d x)}{15 d}+\frac{\left(12 a^3 A b+24 a^2 b^2 B+3 a^4 B+16 a A b^3+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(8 a^2 A+25 a b B+18 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a \left(60 a^2 A b+15 a^3 B+110 a b^2 B+56 A b^3\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{a (5 a B+8 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}","\frac{\left(60 a^2 A b^2+8 a^4 A+40 a^3 b B+60 a b^3 B+15 A b^4\right) \tan (c+d x)}{15 d}+\frac{\left(12 a^3 A b+24 a^2 b^2 B+3 a^4 B+16 a A b^3+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(8 a^2 A+25 a b B+18 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a \left(60 a^2 A b+15 a^3 B+110 a b^2 B+56 A b^3\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{a (5 a B+8 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"((12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*Tan[c + d*x])/(15*d) + (a*(60*a^2*A*b + 56*A*b^3 + 15*a^3*B + 110*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a^2*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (a*(8*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*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,31,0.2581,1,"{2989, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
249,1,324,0,0.8020344,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]","\frac{\left(32 a^3 A b+60 a^2 b^2 B+8 a^4 B+40 a A b^3+15 b^4 B\right) \tan (c+d x)}{15 d}+\frac{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left(25 a^2 A+72 a b B+48 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{a \left(16 a^2 A b+4 a^3 B+27 a b^2 B+13 A b^3\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a (2 a B+3 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}","\frac{\left(32 a^3 A b+60 a^2 b^2 B+8 a^4 B+40 a A b^3+15 b^4 B\right) \tan (c+d x)}{15 d}+\frac{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left(25 a^2 A+72 a b B+48 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{a \left(16 a^2 A b+4 a^3 B+27 a b^2 B+13 A b^3\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a (2 a B+3 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]])/(16*d) + ((32*a^3*A*b + 40*a*A*b^3 + 8*a^4*B + 60*a^2*b^2*B + 15*b^4*B)*Tan[c + d*x])/(15*d) + ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a*(16*a^2*A*b + 13*A*b^3 + 4*a^3*B + 27*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*(3*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*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",9,9,31,0.2903,1,"{2989, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
250,1,178,0,0.4931826,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(2 a^2+b^2\right) (A b-a B)}{2 b^4}+\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 b d}","-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \sin (c+d x)}{3 b^3 d}-\frac{2 a^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(2 a^2+b^2\right) (A b-a B)}{2 b^4}+\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"((2*a^2 + b^2)*(A*b - a*B)*x)/(2*b^4) - (2*a^3*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*Sin[c + d*x])/(3*b^3*d) + ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",6,6,31,0.1935,1,"{2990, 3049, 3023, 2735, 2659, 205}"
251,1,134,0,0.287214,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 a^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}-\frac{x \left(-2 a^2 B+2 a A b-b^2 B\right)}{2 b^3}+\frac{(A b-a B) \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}","\frac{2 a^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}-\frac{x \left(-2 a^2 B+2 a A b-b^2 B\right)}{2 b^3}+\frac{(A b-a B) \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}",1,"-((2*a*A*b - 2*a^2*B - b^2*B)*x)/(2*b^3) + (2*a^2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((A*b - a*B)*Sin[c + d*x])/(b^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,31,0.1613,1,"{2990, 3023, 2735, 2659, 205}"
252,1,89,0,0.1740943,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","-\frac{2 a (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x (A b-a B)}{b^2}+\frac{B \sin (c+d x)}{b d}","-\frac{2 a (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x (A b-a B)}{b^2}+\frac{B \sin (c+d x)}{b d}",1,"((A*b - a*B)*x)/b^2 - (2*a*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(b*d)","A",6,6,29,0.2069,1,"{2968, 3023, 12, 2735, 2659, 205}"
253,1,67,0,0.0777636,"\int \frac{A+B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{2 (A b-a B) \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{B x}{b}","\frac{2 (A b-a B) \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{B x}{b}",1,"(B*x)/b + (2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",3,3,23,0.1304,1,"{2735, 2659, 205}"
254,1,76,0,0.1150536,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (A b-a B) \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{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (A b-a B) \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*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + (A*ArcTanh[Sin[c + d*x]])/(a*d)","A",4,4,29,0.1379,1,"{3001, 3770, 2659, 205}"
255,1,99,0,0.1832283,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 b (A b-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 \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 b (A b-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 \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*b*(A*b - a*B)*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",6,6,31,0.1935,1,"{3000, 12, 2747, 3770, 2659, 205}"
256,1,143,0,0.4903367,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","-\frac{2 b^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{\left(a^2 A-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^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{\left(a^2 A-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^2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) + ((a^2*A + 2*A*b^2 - 2*a*b*B)*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",6,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
257,1,187,0,0.7698693,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{2 b^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{\left(2 a^2 A-3 a b B+3 A b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{\left(a^2+2 b^2\right) (A b-a B) \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^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{\left(2 a^2 A-3 a b B+3 A b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{\left(a^2+2 b^2\right) (A b-a B) \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^3*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((a^2 + 2*b^2)*(A*b - a*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B)*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",7,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
258,1,263,0,0.6589143,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(2 a^2 A b-3 a^3 B+2 a b^2 B-A b^3\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b-3 a^3 B+4 a b^2 B-3 A b^3\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 (A b-a B) \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 B+2 a A b+b^2 B\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 B+4 a A b-b^2 B\right)}{2 b^4}","\frac{\left(2 a^2 A b-3 a^3 B+2 a b^2 B-A b^3\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2 A b-3 a^3 B+4 a b^2 B-3 A b^3\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 (A b-a B) \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 B+2 a A b+b^2 B\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 B+4 a A b-b^2 B\right)}{2 b^4}",1,"-((4*a*A*b - 6*a^2*B - b^2*B)*x)/(2*b^4) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*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*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{2989, 3049, 3023, 2735, 2659, 205}"
259,1,155,0,0.4401183,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 a \left(a^2 A b-2 a^3 B+3 a b^2 B-2 A b^3\right) \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 (A b-a B) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (A b-2 a B)}{b^3}+\frac{B \sin (c+d x)}{b^2 d}","-\frac{2 a \left(a^2 A b-2 a^3 B+3 a b^2 B-2 A b^3\right) \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 (A b-a B) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x (A b-2 a B)}{b^3}+\frac{B \sin (c+d x)}{b^2 d}",1,"((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (B*Sin[c + d*x])/(b^2*d) - (a^2*(A*b - a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,31,0.1613,1,"{2988, 3023, 2735, 2659, 205}"
260,1,122,0,0.240552,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(a^3 B-2 a b^2 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)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B x}{b^2}","-\frac{2 \left(a^3 B-2 a b^2 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)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B x}{b^2}",1,"(B*x)/b^2 - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,29,0.1724,1,"{2968, 3021, 2735, 2659, 205}"
261,1,100,0,0.0885387,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{2 (a A-b B) \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{(A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 (a A-b B) \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{(A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*(a*A - b*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,23,0.1739,1,"{2754, 12, 2659, 205}"
262,1,133,0,0.2816643,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","-\frac{2 \left(2 a^2 A b+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{b (A b-a B) \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 \left(2 a^2 A b+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{b (A b-a B) \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*(2*a^2*A*b - A*b^3 - a^3*B)*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) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,29,0.1724,1,"{3000, 3001, 3770, 2659, 205}"
263,1,189,0,0.6731644,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b \left(3 a^2 A b-2 a^3 B+a b^2 B-2 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^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 A+a b B-2 A b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \tan (c+d x)}{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 b \left(3 a^2 A b-2 a^3 B+a b^2 B-2 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^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 A+a b B-2 A b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \tan (c+d x)}{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*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a*b^2*B)*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) + ((a^2*A - 2*A*b^2 + a*b*B)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
264,1,270,0,0.9754677,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b^2 \left(4 a^2 A b-3 a^3 B+2 a b^2 B-3 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^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(2 a^2 A b+a^3 (-B)+2 a b^2 B-3 A b^3\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 A-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(a^2 A+2 a b B-3 A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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^2 \left(4 a^2 A b-3 a^3 B+2 a b^2 B-3 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^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(2 a^2 A b+a^3 (-B)+2 a b^2 B-3 A b^3\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2 A-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(a^2 A+2 a b B-3 A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2*A + 6*A*b^2 - 4*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B)*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
265,1,398,0,1.7236463,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-6 a b^4 B+2 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-20 a b^4 B+12 A b^5\right) \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 (A b-a B) \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 A b-4 a^3 B+7 a b^2 B-5 A b^3\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(3 a^3 A b+10 a^2 b^2 B-6 a^4 B-6 a A b^3-b^4 B\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 B+6 a A b-b^2 B\right)}{2 b^5}","\frac{\left(-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-6 a b^4 B+2 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-20 a b^4 B+12 A b^5\right) \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 (A b-a B) \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 A b-4 a^3 B+7 a b^2 B-5 A b^3\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(3 a^3 A b+10 a^2 b^2 B-6 a^4 B-6 a A b^3-b^4 B\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 B+6 a A b-b^2 B\right)}{2 b^5}",1,"-((6*a*A*b - 12*a^2*B - b^2*B)*x)/(2*b^5) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*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*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*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*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*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",7,7,31,0.2258,1,"{2989, 3047, 3049, 3023, 2735, 2659, 205}"
266,1,280,0,1.2218652,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^2 A b^3+2 a^4 A b+15 a^3 b^2 B-6 a^5 B-12 a b^4 B+6 A b^5\right) \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 (A b-a B) \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 A b-3 a^3 B+6 a b^2 B-4 A b^3\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (A b-3 a B)}{b^4}","-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^2 A b^3+2 a^4 A b+15 a^3 b^2 B-6 a^5 B-12 a b^4 B+6 A b^5\right) \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 (A b-a B) \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 A b-3 a^3 B+6 a b^2 B-4 A b^3\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{x (A b-3 a B)}{b^4}",1,"((A*b - 3*a*B)*x)/b^4 - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(A*b - a*B)*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*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{2989, 3031, 3023, 2735, 2659, 205}"
267,1,211,0,0.5644177,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2 A b^3+5 a^3 b^2 B-2 a^5 B-6 a b^4 B+2 A b^5\right) \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 (A b-a B) \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 A b-3 a^3 B+6 a b^2 B-4 A b^3\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{B x}{b^3}","\frac{\left(a^2 A b^3+5 a^3 b^2 B-2 a^5 B-6 a b^4 B+2 A b^5\right) \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 (A b-a B) \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 A b-3 a^3 B+6 a b^2 B-4 A b^3\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{B x}{b^3}",1,"(B*x)/b^3 + ((a^2*A*b^3 + 2*A*b^5 - 2*a^5*B + 5*a^3*b^2*B - 6*a*b^4*B)*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*(A*b - a*B)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*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,"{2988, 3021, 2735, 2659, 205}"
268,1,180,0,0.2900818,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(a^2 (-B)+3 a A b-2 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 A b+a^3 B-4 a b^2 B+2 A b^3\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(a^2 (-B)+3 a A b-2 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 A b+a^3 B-4 a b^2 B+2 A b^3\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-(((3*a*A*b - a^2*B - 2*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,29,0.2069,1,"{2968, 3021, 2754, 12, 2659, 205}"
269,1,164,0,0.1881677,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\left(2 a^2 A-3 a b B+A 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{\left(a^2 (-B)+3 a A b-2 b^2 B\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(A b-a B) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(2 a^2 A-3 a b B+A 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{\left(a^2 (-B)+3 a A b-2 b^2 B\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(A b-a B) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,4,23,0.1739,1,"{2754, 12, 2659, 205}"
270,1,214,0,0.7057147,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-5 a^2 A b^3+6 a^4 A b-a^3 b^2 B-2 a^5 B+2 A b^5\right) \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 A b-3 a^3 B-2 A b^3\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \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{\left(-5 a^2 A b^3+6 a^4 A b-a^3 b^2 B-2 a^5 B+2 A b^5\right) \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 A b-3 a^3 B-2 A b^3\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \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,"-(((6*a^4*A*b - 5*a^2*A*b^3 + 2*A*b^5 - 2*a^5*B - a^3*b^2*B)*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) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,29,0.2069,1,"{3000, 3055, 3001, 3770, 2659, 205}"
271,1,299,0,1.7644528,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{b \left(-15 a^2 A b^3+12 a^4 A b+5 a^3 b^2 B-6 a^5 B-2 a b^4 B+6 A b^5\right) \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+2 a^4 A+5 a^3 b B-2 a b^3 B+6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(6 a^2 A b-4 a^3 B+a b^2 B-3 A b^3\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \tan (c+d x)}{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{b \left(-15 a^2 A b^3+12 a^4 A b+5 a^3 b^2 B-6 a^5 B-2 a b^4 B+6 A b^5\right) \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+2 a^4 A+5 a^3 b B-2 a b^3 B+6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(6 a^2 A b-4 a^3 B+a b^2 B-3 A b^3\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \tan (c+d x)}{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,"(b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*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) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(6*a^2*A*b - 3*A*b^3 - 4*a^3*B + a*b^2*B)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
272,1,402,0,2.2364491,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","-\frac{b^2 \left(-29 a^2 A b^3+20 a^4 A b+15 a^3 b^2 B-12 a^5 B-6 a b^4 B+12 A b^5\right) \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{\left(-21 a^2 A b^3+6 a^4 A b+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 A-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-10 a^2 A b^2+a^4 A+6 a^3 b B-3 a b^3 B+6 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(7 a^2 A b-5 a^3 B+2 a b^2 B-4 A b^3\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{b (A b-a B) \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^2 \left(-29 a^2 A b^3+20 a^4 A b+15 a^3 b^2 B-12 a^5 B-6 a b^4 B+12 A b^5\right) \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{\left(-21 a^2 A b^3+6 a^4 A b+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2 A-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-10 a^2 A b^2+a^4 A+6 a^3 b B-3 a b^3 B+6 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(7 a^2 A b-5 a^3 B+2 a b^2 B-4 A b^3\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{b (A b-a B) \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^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((a^2*A + 12*A*b^2 - 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
273,1,409,0,5.1748046,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(3 a^3 A b+23 a^2 b^2 B-12 a^4 B-8 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+20 a b^6 B-8 A b^7\right) \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{a (A b-a B) \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{a \left(a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\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^2 \left(-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-12 a b^4 B+6 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x (A b-4 a B)}{b^5}","-\frac{\left(3 a^3 A b+23 a^2 b^2 B-12 a^4 B-8 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+20 a b^6 B-8 A b^7\right) \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{a (A b-a B) \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{a \left(a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\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^2 \left(-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-12 a b^4 B+6 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x (A b-4 a B)}{b^5}",1,"((A*b - 4*a*B)*x)/b^5 - (a*(2*a^6*A*b - 7*a^4*A*b^3 + 8*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 28*a^5*b^2*B - 35*a^3*b^4*B + 20*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((3*a^3*A*b - 8*a*A*b^3 - 12*a^4*B + 23*a^2*b^2*B - 6*b^4*B)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*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^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,7,31,0.2258,1,"{2989, 3047, 3031, 3023, 2735, 2659, 205}"
274,1,301,0,1.2066845,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(3 a^2 A b^5-7 a^5 b^2 B+8 a^3 b^4 B+2 a^7 B-8 a b^6 B+2 A b^7\right) \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{a (A b-a B) \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{a^2 \left(3 a^3 B-8 a b^2 B+5 A b^3\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a \left(a^2 A b^3-28 a^3 b^2 B+9 a^5 B+34 a b^4 B-16 A b^5\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{B x}{b^4}","-\frac{\left(3 a^2 A b^5-7 a^5 b^2 B+8 a^3 b^4 B+2 a^7 B-8 a b^6 B+2 A b^7\right) \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{a (A b-a B) \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{a^2 \left(3 a^3 B-8 a b^2 B+5 A b^3\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a \left(a^2 A b^3-28 a^3 b^2 B+9 a^5 B+34 a b^4 B-16 A b^5\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{B x}{b^4}",1,"(B*x)/b^4 - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) + (a*(A*b - a*B)*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*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 34*a*b^4*B)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,31,0.1935,1,"{2989, 3031, 3021, 2735, 2659, 205}"
275,1,274,0,0.6360467,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","\frac{\left(a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right) \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^2 (A b-a B) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 A b^3+a^4 A b-5 a^3 b^2 B+2 a^5 B+18 a b^4 B-6 A b^5\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}","\frac{\left(a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 (A b-a B) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 A b^3+a^4 A b-5 a^3 b^2 B+2 a^5 B+18 a b^4 B-6 A b^5\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*(A*b - a*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*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,"{2988, 3021, 2754, 12, 2659, 205}"
276,1,263,0,0.5301786,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(4 a^2 A b+a^3 (-B)-4 a b^2 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)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(2 a^3 A b-10 a^2 b^2 B+a^4 B+13 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(2 a^2 A b+a^3 B-6 a b^2 B+3 A b^3\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","-\frac{\left(4 a^2 A b+a^3 (-B)-4 a b^2 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)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(2 a^3 A b-10 a^2 b^2 B+a^4 B+13 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(2 a^2 A b+a^3 B-6 a b^2 B+3 A b^3\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"-(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,6,29,0.2069,1,"{2968, 3021, 2754, 12, 2659, 205}"
277,1,237,0,0.4776344,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4,x]","\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right) \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(11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{(A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^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)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{(A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,4,23,0.1739,1,"{2754, 12, 2659, 205}"
278,1,301,0,1.5088636,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","-\frac{\left(-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \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{b \left(-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+6 A b^5\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b \left(8 a^2 A b-5 a^3 B-3 A b^3\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{\left(-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right) \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{b \left(-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+6 A b^5\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b \left(8 a^2 A b-5 a^3 B-3 A b^3\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-(((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*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) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,6,29,0.2069,1,"{3000, 3055, 3001, 3770, 2659, 205}"
279,1,420,0,6.2209646,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{b \left(-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right) \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(-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b \left(9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}","\frac{b \left(-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right) \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(-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b \left(9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}",1,"(b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*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) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
280,1,547,0,7.3044603,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","-\frac{b^2 \left(-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+8 a b^6 B-20 A b^7\right) \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)^{7/2} (a+b)^{7/2}}-\frac{\left(-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 A-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left(-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+4 a b^5 B-10 A b^6\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-8 a b^4 B+20 A b^5\right) \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{b \left(10 a^2 A b-7 a^3 B+2 a b^2 B-5 A b^3\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{b (A b-a B) \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{b^2 \left(-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+8 a b^6 B-20 A b^7\right) \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)^{7/2} (a+b)^{7/2}}-\frac{\left(-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(a^2 A-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left(-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+4 a b^5 B-10 A b^6\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-8 a b^4 B+20 A b^5\right) \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{b \left(10 a^2 A b-7 a^3 B+2 a b^2 B-5 A b^3\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{b (A b-a B) \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,"-((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ((a^2*A + 20*A*b^2 - 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",9,6,31,0.1935,1,"{3000, 3055, 3001, 3770, 2659, 205}"
281,1,28,0,0.015511,"\int \frac{\cos ^3(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{B \sin (c+d x)}{d}-\frac{B \sin ^3(c+d x)}{3 d}","\frac{B \sin (c+d x)}{d}-\frac{B \sin ^3(c+d x)}{3 d}",1,"(B*Sin[c + d*x])/d - (B*Sin[c + d*x]^3)/(3*d)","A",3,2,34,0.05882,1,"{21, 2633}"
282,1,27,0,0.0145795,"\int \frac{\cos ^2(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}",1,"(B*x)/2 + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",3,3,34,0.08824,1,"{21, 2635, 8}"
283,1,11,0,0.0087709,"\int \frac{\cos (c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{B \sin (c+d x)}{d}","\frac{B \sin (c+d x)}{d}",1,"(B*Sin[c + d*x])/d","A",2,2,32,0.06250,1,"{21, 2637}"
284,1,3,0,0.0011028,"\int \frac{a B+b B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","B x","B x",1,"B*x","A",2,2,26,0.07692,1,"{21, 8}"
285,1,12,0,0.0068244,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/d","A",2,2,32,0.06250,1,"{21, 3770}"
286,1,11,0,0.011593,"\int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{B \tan (c+d x)}{d}","\frac{B \tan (c+d x)}{d}",1,"(B*Tan[c + d*x])/d","A",3,3,34,0.08824,1,"{21, 3767, 8}"
287,1,36,0,0.0197869,"\int \frac{(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 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) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",3,3,34,0.08824,1,"{21, 3768, 3770}"
288,1,28,0,0.0155774,"\int \frac{(a B+b B \cos (c+d x)) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}","\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}",1,"(B*Tan[c + d*x])/d + (B*Tan[c + d*x]^3)/(3*d)","A",3,2,34,0.05882,1,"{21, 3767}"
289,1,114,0,0.2165498,"\int \frac{\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 a^3 B \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{B x \left(2 a^2+b^2\right)}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{2 a^3 B \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{B x \left(2 a^2+b^2\right)}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}",1,"((2*a^2 + b^2)*B*x)/(2*b^3) - (2*a^3*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*B*Sin[c + d*x])/(b^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",6,6,34,0.1765,1,"{21, 2793, 3023, 2735, 2659, 205}"
290,1,79,0,0.1316424,"\int \frac{\cos ^2(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{2 a^2 B \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 B x}{b^2}+\frac{B \sin (c+d x)}{b d}","\frac{2 a^2 B \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 B x}{b^2}+\frac{B \sin (c+d x)}{b d}",1,"-((a*B*x)/b^2) + (2*a^2*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(b*d)","A",6,6,34,0.1765,1,"{21, 2746, 12, 2735, 2659, 205}"
291,1,61,0,0.0666067,"\int \frac{\cos (c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{B x}{b}-\frac{2 a B \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{B x}{b}-\frac{2 a B \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}}",1,"(B*x)/b - (2*a*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",4,4,32,0.1250,1,"{21, 2735, 2659, 205}"
292,1,50,0,0.0353222,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[(a*B + b*B*Cos[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)}{d \sqrt{a-b} \sqrt{a+b}}","\frac{2 B \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}}",1,"(2*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",3,3,26,0.1154,1,"{21, 2659, 205}"
293,1,70,0,0.0805626,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 b B \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 \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*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",5,5,32,0.1562,1,"{21, 2747, 3770, 2659, 205}"
294,1,88,0,0.1448057,"\int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 b^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{b B \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}","\frac{2 b^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{b B \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}",1,"(2*b^2*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (b*B*ArcTanh[Sin[c + d*x]])/(a^2*d) + (B*Tan[c + d*x])/(a*d)","A",7,7,34,0.2059,1,"{21, 2802, 12, 2747, 3770, 2659, 205}"
295,1,123,0,0.3460146,"\int \frac{(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b^3 B \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{B \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{b B \tan (c+d x)}{a^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 b^3 B \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{B \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{b B \tan (c+d x)}{a^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(-2*b^3*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) + ((a^2 + 2*b^2)*B*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*B*Tan[c + d*x])/(a^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",7,7,34,0.2059,1,"{21, 2802, 3055, 3001, 3770, 2659, 205}"
296,1,386,0,0.7854794,"\int \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(-24 a^2 B+36 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \left(24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(24 a^2 A b-16 a^3 B-36 a b^2 B+75 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)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(24 a^3 A b-24 a^2 b^2 B-16 a^4 B+57 a A b^3+147 b^4 B\right) \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 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 B \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 B+36 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \left(24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(24 a^2 A b-16 a^3 B-36 a b^2 B+75 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)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(24 a^3 A b-24 a^2 b^2 B-16 a^4 B+57 a A b^3+147 b^4 B\right) \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 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 B \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*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*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*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*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*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) - (2*(36*a*A*b - 24*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) + (2*(3*A*b - 2*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*B*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)","A",9,9,33,0.2727,1,"{2990, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
297,1,303,0,0.5415723,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(-8 a^2 B+14 a A b-25 b^2 B\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 B+14 a A b-25 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)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 A b-8 a^3 B-19 a b^2 B-63 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)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 B \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 B+14 a A b-25 b^2 B\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 B+14 a A b-25 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)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(14 a^2 A b-8 a^3 B-19 a b^2 B-63 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)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(-2*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*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*A*b - 8*a^2*B - 25*b^2*B)*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*A*b - 8*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*B*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,"{2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
298,1,231,0,0.4075887,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right) (5 A b-2 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)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 B+5 a A b+9 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)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 A b-2 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}","-\frac{2 \left(a^2-b^2\right) (5 A b-2 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)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 B+5 a A b+9 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)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 A b-2 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*(5*a*A*b - 2*a^2*B + 9*b^2*B)*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*A*b - 2*a*B)*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*A*b - 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",8,8,31,0.2581,1,"{2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
299,1,171,0,0.2196545,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","-\frac{2 B \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 B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","-\frac{2 B \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 B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*A*b + a*B)*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)*B*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*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
300,1,178,0,0.360206,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 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)}{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 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{2 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)}{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 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,"(2*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)]) + (2*A*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*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",8,8,31,0.2581,1,"{3002, 2655, 2653, 2803, 2663, 2661, 2807, 2805}"
301,1,213,0,0.6067323,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{A \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 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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{A \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,"-((A*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",9,9,33,0.2727,1,"{2999, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
302,1,292,0,0.9547918,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\left(4 a^2 A+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+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a B+3 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 d \sqrt{a+b \cos (c+d x)}}-\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+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+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a B+3 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 d \sqrt{a+b \cos (c+d x)}}-\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)*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*A - A*b^2 + 4*a*b*B)*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",10,10,33,0.3030,1,"{2999, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
303,1,378,0,1.3386235,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\left(16 a^2 A+6 a b B-3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}+\frac{\left(16 a^2 A+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(16 a^2 A+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 A b+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{\left(16 a^2 A+6 a b B-3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}+\frac{\left(16 a^2 A+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(16 a^2 A+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 A b+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,"-((16*a^2*A - 3*A*b^2 + 6*a*b*B)*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)]) + ((16*a^2*A - A*b^2 + 18*a*b*B)*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]]) + ((4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*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]]) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*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",11,10,33,0.3030,1,"{2999, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
304,1,378,0,0.7330614,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(-8 a^2 B+18 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right) \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 A b-8 a^3 B-39 a b^2 B-75 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)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right) \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 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \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 B+18 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right) \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 A b-8 a^3 B-39 a b^2 B-75 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)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right) \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 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(-2*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*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*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*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*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) - (2*(18*a*A*b - 8*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*B*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,"{2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
305,1,297,0,0.527524,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 \left(-6 a^2 B+21 a A b+25 b^2 B\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 B+21 a A b+25 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)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 A b-6 a^3 B+82 a b^2 B+63 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)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}","\frac{2 \left(-6 a^2 B+21 a A b+25 b^2 B\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 B+21 a A b+25 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)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(21 a^2 A b-6 a^3 B+82 a b^2 B+63 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)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}",1,"(2*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*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*A*b - 6*a^2*B + 25*b^2*B)*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*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)","A",9,8,31,0.2581,1,"{2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
306,1,225,0,0.3522475,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right) (3 a B+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)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 B+20 a A b+9 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)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","-\frac{2 \left(a^2-b^2\right) (3 a B+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)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 B+20 a A b+9 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)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(20*a*A*b + 3*a^2*B + 9*b^2*B)*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*A*b + 3*a*B)*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*A*b + 3*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",7,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
307,1,236,0,0.7076822,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 \left(a^2 (-B)+3 a A b+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 d \sqrt{a+b \cos (c+d x)}}+\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 (4 a B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{2 \left(a^2 (-B)+3 a A b+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 d \sqrt{a+b \cos (c+d x)}}+\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 (4 a B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(3*A*b + 4*a*B)*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*A*b - a^2*B + b^2*B)*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*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*b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,31,0.2903,1,"{2990, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
308,1,232,0,0.6855516,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\left(a^2 A+2 a b B+2 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)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a A-2 b 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 (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{a A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}","\frac{\left(a^2 A+2 a b B+2 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)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a A-2 b 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 (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{a A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"-(((a*A - 2*b*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^2*A + 2*A*b^2 + 2*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]]) + (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]]) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",9,9,33,0.2727,1,"{2989, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
309,1,295,0,1.0580667,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\left(4 a^2 B+7 a A b+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+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) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a B+5 A 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 A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","\frac{\left(4 a^2 B+7 a A b+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+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) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a B+5 A 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 A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-((5*A*b + 4*a*B)*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*A*b + 4*a^2*B + 8*b^2*B)*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*A + 3*A*b^2 + 12*a*b*B)*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*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,33,0.3030,1,"{2989, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
310,1,375,0,1.4499379,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\left(16 a^2 A+30 a b B+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 A+42 a b B+17 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 A+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 A b+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{(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}","\frac{\left(16 a^2 A+30 a b B+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 A+42 a b B+17 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 A+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 A b+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{(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"-((16*a^2*A + 3*A*b^2 + 30*a*b*B)*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*A + 17*A*b^2 + 42*a*b*B)*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*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*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*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((7*A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,33,0.3030,1,"{2989, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
311,1,462,0,0.931401,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","-\frac{2 \left(-8 a^2 B+22 a A b-81 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(110 a^2 A b-40 a^3 B-335 a b^2 B-539 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right) \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 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-3705 a b^4 B-1617 A b^5\right) \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 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \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 B+22 a A b-81 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(110 a^2 A b-40 a^3 B-335 a b^2 B-539 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right) \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 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-3705 a b^4 B-1617 A b^5\right) \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 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(-2*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*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*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*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*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B - 81*b^2*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*B*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,"{2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
312,1,372,0,0.7963672,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{2 \left(-10 a^2 B+45 a A b+49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right) \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 A b-10 a^3 B+114 a b^2 B+75 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)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(45 a^3 A b+279 a^2 b^2 B-10 a^4 B+435 a A b^3+147 b^4 B\right) \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 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}","\frac{2 \left(-10 a^2 B+45 a A b+49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(45 a^2 A b-10 a^3 B+114 a b^2 B+75 A b^3\right) \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 A b-10 a^3 B+114 a b^2 B+75 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)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(45 a^3 A b+279 a^2 b^2 B-10 a^4 B+435 a A b^3+147 b^4 B\right) \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 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}",1,"(2*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*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*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*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*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)","A",10,8,31,0.2581,1,"{2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
313,1,288,0,0.5163004,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{2 \left(15 a^2 B+56 a A b+25 b^2 B\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 B+56 a A b+25 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)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 A b+15 a^3 B+145 a b^2 B+63 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)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}","\frac{2 \left(15 a^2 B+56 a A b+25 b^2 B\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 B+56 a A b+25 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)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(161 a^2 A b+15 a^3 B+145 a b^2 B+63 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)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*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*A*b + 15*a^2*B + 25*b^2*B)*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*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A*b + 5*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",8,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
314,1,292,0,1.0136668,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 \left(10 a^2 A b-8 a^3 B+8 a b^2 B+5 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)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2 B+35 a A b+9 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)}{15 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 b (8 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}","\frac{2 \left(10 a^2 A b-8 a^3 B+8 a b^2 B+5 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)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2 B+35 a A b+9 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)}{15 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 b (8 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(35*a*A*b + 23*a^2*B + 9*b^2*B)*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*A*b + 5*A*b^3 - 8*a^3*B + 8*a*b^2*B)*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*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*b*(5*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",10,10,31,0.3226,1,"{2990, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
315,1,296,0,1.1090554,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\left(3 a^3 A+4 a^2 b B+12 a A b^2+2 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 A-14 a b B-6 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)}{3 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 (3 a A-2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}","\frac{\left(3 a^3 A+4 a^2 b B+12 a A b^2+2 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 A-14 a b B-6 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)}{3 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 (3 a A-2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"-((3*a^2*A - 6*A*b^2 - 14*a*b*B)*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*A + 12*a*A*b^2 + 4*a^2*b*B + 2*b^3*B)*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*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*A - 2*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d","A",10,10,33,0.3030,1,"{2989, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
316,1,315,0,1.0552851,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\left(11 a^2 A b+4 a^3 B+16 a b^2 B+8 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)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 B+9 a A b-8 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)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 A+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{a (4 a B+7 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}","\frac{\left(11 a^2 A b+4 a^3 B+16 a b^2 B+8 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)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^2 B+9 a A b-8 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)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 A+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{a (4 a B+7 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"-((9*a*A*b + 4*a^2*B - 8*b^2*B)*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*A*b + 8*A*b^3 + 4*a^3*B + 16*a*b^2*B)*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*A + 15*A*b^2 + 20*a*b*B)*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*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,33,0.3030,1,"{2989, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
317,1,376,0,1.4331102,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\left(16 a^2 A+54 a b B+33 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(16 a^3 A+66 a^2 b B+59 a A b^2+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(16 a^2 A+54 a b B+33 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 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 A b+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{a (2 a B+3 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\left(16 a^2 A+54 a b B+33 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\left(16 a^3 A+66 a^2 b B+59 a A b^2+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(16 a^2 A+54 a b B+33 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 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(20 a^2 A b+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{a (2 a B+3 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"-((16*a^2*A + 33*A*b^2 + 54*a*b*B)*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*A + 59*a*A*b^2 + 66*a^2*b*B + 48*b^3*B)*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*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*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*A + 33*A*b^2 + 54*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(3*A*b + 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,11,33,0.3333,1,"{2989, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
318,1,465,0,1.8453724,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{\left(284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(356 a^2 A b+128 a^3 B+472 a b^2 B+133 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(284 a^2 A b+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 A b^2+48 a^4 A+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{\left(36 a^2 A+104 a b B+59 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}","\frac{\left(284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(356 a^2 A b+128 a^3 B+472 a b^2 B+133 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(284 a^2 A b+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 A b^2+48 a^4 A+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{\left(36 a^2 A+104 a b B+59 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"-((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*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*A*b + 133*A*b^3 + 128*a^3*B + 472*a*b^2*B)*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*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*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*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*d) + (a*(11*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*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,11,33,0.3333,1,"{2989, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
319,1,320,0,0.6190891,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \left(-24 a^2 B+28 a A b-25 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}-\frac{2 \left(56 a^3 A b-32 a^2 b^2 B-48 a^4 B+49 a A b^3-25 b^4 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)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(56 a^2 A b-48 a^3 B-44 a b^2 B+63 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)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 B \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 B+28 a A b-25 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}-\frac{2 \left(56 a^3 A b-32 a^2 b^2 B-48 a^4 B+49 a A b^3-25 b^4 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)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(56 a^2 A b-48 a^3 B-44 a b^2 B+63 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)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 B \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}",1,"(2*(56*a^2*A*b + 63*A*b^3 - 48*a^3*B - 44*a*b^2*B)*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*A*b + 49*a*A*b^3 - 48*a^4*B - 32*a^2*b^2*B - 25*b^4*B)*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*(28*a*A*b - 24*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*B*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)","A",8,8,33,0.2424,1,"{2990, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
320,1,246,0,0.4296678,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(10 a^2 A b-8 a^3 B-7 a b^2 B+5 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)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 B+10 a A b-9 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)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 A b-4 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}","\frac{2 \left(10 a^2 A b-8 a^3 B-7 a b^2 B+5 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)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 B+10 a A b-9 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)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 A b-4 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(-2*(10*a*A*b - 8*a^2*B - 9*b^2*B)*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*A*b + 5*A*b^3 - 8*a^3*B - 7*a*b^2*B)*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*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*B*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,"{2990, 3023, 2752, 2663, 2661, 2655, 2653}"
321,1,183,0,0.2917973,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \left(-2 a^2 B+3 a A b-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 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 A b-2 a B) \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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","-\frac{2 \left(-2 a^2 B+3 a A b-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 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 A b-2 a B) \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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(2*(3*A*b - 2*a*B)*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*A*b - 2*a^2*B - b^2*B)*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*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",7,7,31,0.2258,1,"{2968, 3023, 2752, 2663, 2661, 2655, 2653}"
322,1,130,0,0.1262017,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 (A b-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)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 B \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 b-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)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 B \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*B*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*b - a*B)*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",5,5,25,0.2000,1,"{2752, 2663, 2661, 2655, 2653}"
323,1,118,0,0.3128657,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*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 \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 \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 \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*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*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",5,5,31,0.1613,1,"{3002, 2663, 2661, 2807, 2805}"
324,1,216,0,0.6551182,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\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)}{d \sqrt{a+b \cos (c+d x)}}-\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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\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)}{d \sqrt{a+b \cos (c+d x)}}-\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*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",9,9,33,0.2727,1,"{3000, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
325,1,299,0,0.9549726,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(4 a^2 A-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-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]]) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B)*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",10,10,33,0.3030,1,"{3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
326,1,387,0,0.7271219,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (A b-a B) \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 B+5 a A b+b^2 B\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 A b-24 a^3 B+9 a b^2 B-5 A b^3\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 A b-48 a^3 B-12 a b^2 B+5 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)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(40 a^3 A b+24 a^2 b^2 B-48 a^4 B-25 a A b^3+9 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)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 a (A b-a B) \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 B+5 a A b+b^2 B\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 A b-24 a^3 B+9 a b^2 B-5 A b^3\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 A b-48 a^3 B-12 a b^2 B+5 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)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(40 a^3 A b+24 a^2 b^2 B-48 a^4 B-25 a A b^3+9 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)}{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*A*b - 25*a*A*b^3 - 48*a^4*B + 24*a^2*b^2*B + 9*b^4*B)*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*A*b + 5*A*b^3 - 48*a^3*B - 12*a*b^2*B)*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*(A*b - a*B)*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*A*b - 5*A*b^3 - 24*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*A*b - 6*a^2*B + b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",8,8,33,0.2424,1,"{2989, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
327,1,262,0,0.4862154,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 (A b-a B) \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 B+6 a A b-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 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 A b-8 a^3 B+5 a b^2 B-3 A b^3\right) \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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}","-\frac{2 a^2 (A b-a B) \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 B+6 a A b-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 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 A b-8 a^3 B+5 a b^2 B-3 A b^3\right) \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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*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*A*b - 8*a^2*B - b^2*B)*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*(A*b - a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)","A",7,7,33,0.2121,1,"{2988, 3023, 2752, 2663, 2661, 2655, 2653}"
328,1,204,0,0.3317669,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 B+a A b+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)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (A b-2 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)}{b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a (A b-a B) \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 B+a A b+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)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (A b-2 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)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(a*A*b - 2*a^2*B + b^2*B)*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*(A*b - 2*a*B)*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*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,31,0.2258,1,"{2968, 3021, 2752, 2663, 2661, 2655, 2653}"
329,1,185,0,0.2305435,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{b 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}} 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 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{b 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}} 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 - a*B)*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*B*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*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
330,1,190,0,0.5084499,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (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)}{a 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 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (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)}{a 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)}}",1,"(-2*(A*b - a*B)*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*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*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,31,0.2258,1,"{3000, 3059, 2655, 2653, 12, 2807, 2805}"
331,1,303,0,0.9909684,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b \left(a^2 A+2 a b B-3 A b^2\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 A+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 \left(a^2 A+2 a b B-3 A b^2\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 A+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,"-(((a^2*A - 3*A*b^2 + 2*a*b*B)*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*(a^2*A - 3*A*b^2 + 2*a*b*B)*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",10,10,33,0.3030,1,"{3000, 3056, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
332,1,398,0,1.4313936,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{b \left(7 a^2 A b-4 a^3 B+12 a b^2 B-15 A b^3\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(7 a^2 A b-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-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 \left(7 a^2 A b-4 a^3 B+12 a b^2 B-15 A b^3\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(7 a^2 A b-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-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,"((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*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]]) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B)*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*(7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*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",11,10,33,0.3030,1,"{3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
333,1,550,0,1.1871433,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 a (A b-a B) \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{2 a \left(5 a^2 A b-8 a^3 B+12 a b^2 B-9 A b^3\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(30 a^3 A b+71 a^2 b^2 B-48 a^4 B-50 a A b^3-3 b^4 B\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{2 \left(-65 a^2 A b^3+40 a^4 A b+98 a^3 b^2 B-64 a^5 B-14 a b^4 B+5 A b^5\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 \left(-80 a^2 A b^3+80 a^4 A b+116 a^3 b^2 B-128 a^5 B+17 a b^4 B-5 A b^5\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(-140 a^3 A b^3+80 a^5 A b+212 a^4 b^2 B-55 a^2 b^4 B-128 a^6 B+40 a A b^5-9 b^6 B\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 a (A b-a B) \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{2 a \left(5 a^2 A b-8 a^3 B+12 a b^2 B-9 A b^3\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(30 a^3 A b+71 a^2 b^2 B-48 a^4 B-50 a A b^3-3 b^4 B\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{2 \left(-65 a^2 A b^3+40 a^4 A b+98 a^3 b^2 B-64 a^5 B-14 a b^4 B+5 A b^5\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 \left(-80 a^2 A b^3+80 a^4 A b+116 a^3 b^2 B-128 a^5 B+17 a b^4 B-5 A b^5\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(-140 a^3 A b^3+80 a^5 A b+212 a^4 b^2 B-55 a^2 b^4 B-128 a^6 B+40 a A b^5-9 b^6 B\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*A*b - 140*a^3*A*b^3 + 40*a*A*b^5 - 128*a^6*B + 212*a^4*b^2*B - 55*a^2*b^4*B - 9*b^6*B)*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*A*b - 80*a^2*A*b^3 - 5*A*b^5 - 128*a^5*B + 116*a^3*b^2*B + 17*a*b^4*B)*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*(A*b - a*B)*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*a*(5*a^2*A*b - 9*A*b^3 - 8*a^3*B + 12*a*b^2*B)*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*A*b - 65*a^2*A*b^3 + 5*A*b^5 - 64*a^5*B + 98*a^3*b^2*B - 14*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*A*b - 50*a*A*b^3 - 48*a^4*B + 71*a^2*b^2*B - 3*b^4*B)*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",9,9,33,0.2727,1,"{2989, 3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
334,1,413,0,0.8046461,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 a (A b-a B) \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 A b-6 a^3 B+10 a b^2 B-7 A b^3\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+a A b+b^2 B\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(8 a^3 A b+16 a^2 b^2 B-16 a^4 B-9 a A b^3+b^4 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^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 A b^3+8 a^4 A b+28 a^3 b^2 B-16 a^5 B-8 a b^4 B+3 A b^5\right) \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 (A b-a B) \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 A b-6 a^3 B+10 a b^2 B-7 A b^3\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+a A b+b^2 B\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(8 a^3 A b+16 a^2 b^2 B-16 a^4 B-9 a A b^3+b^4 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^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 A b^3+8 a^4 A b+28 a^3 b^2 B-16 a^5 B-8 a b^4 B+3 A b^5\right) \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*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Sqrt[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*A*b - 9*a*A*b^3 - 16*a^4*B + 16*a^2*b^2*B + b^4*B)*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*(A*b - a*B)*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*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",8,8,33,0.2424,1,"{2989, 3031, 3023, 2752, 2663, 2661, 2655, 2653}"
335,1,331,0,0.5527173,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 (A b-a B) \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 A b-5 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 A b-8 a^3 B+9 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)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^3 A b+15 a^2 b^2 B-8 a^4 B-6 a A b^3-3 b^4 B\right) \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 (A b-a B) \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 A b-5 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 A b-8 a^3 B+9 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)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^3 A b+15 a^2 b^2 B-8 a^4 B-6 a A b^3-3 b^4 B\right) \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*A*b - 6*a*A*b^3 - 8*a^4*B + 15*a^2*b^2*B - 3*b^4*B)*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*A*b - 3*A*b^3 - 8*a^3*B + 9*a*b^2*B)*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*(A*b - a*B)*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*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,33,0.2121,1,"{2988, 3021, 2752, 2663, 2661, 2655, 2653}"
336,1,307,0,0.4717441,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2 A b+2 a^3 B-6 a b^2 B+3 A b^3\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 B+a A b-3 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 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 A b+2 a^3 B-6 a b^2 B+3 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)}{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 A b+2 a^3 B-6 a b^2 B+3 A b^3\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 (A b-a B) \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 B+a A b-3 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 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 A b+2 a^3 B-6 a b^2 B+3 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)}{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*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*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*A*b + 2*a^2*B - 3*b^2*B)*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*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,31,0.2581,1,"{2968, 3021, 2754, 2752, 2663, 2661, 2655, 2653}"
337,1,275,0,0.3731451,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(a^2 (-B)+4 a A b-3 b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (A b-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)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-B)+4 a A b-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)}{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 (-B)+4 a A b-3 b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (A b-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)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-B)+4 a A b-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)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(4*a*A*b - a^2*B - 3*b^2*B)*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*(A*b - a*B)*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*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
338,1,349,0,1.0990516,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b \left(7 a^2 A b-4 a^3 B-3 A b^3\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 (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (A b-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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(7 a^2 A b-4 a^3 B-3 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)}{3 a^2 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 b \left(7 a^2 A b-4 a^3 B-3 A b^3\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 (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (A b-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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(7 a^2 A b-4 a^3 B-3 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)}{3 a^2 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*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*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*(A*b - a*B)*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*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*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,31,0.3226,1,"{3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
339,1,437,0,1.4809301,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-6 a b^3 B+15 A b^4\right) \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 A+2 a b B-5 A b^2\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 A+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+3 a^4 A+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 \left(-26 a^2 A b^2+3 a^4 A+14 a^3 b B-6 a b^3 B+15 A b^4\right) \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 A+2 a b B-5 A b^2\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 A+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+3 a^4 A+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,"-((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*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*A - 5*A*b^2 + 2*a*b*B)*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*(3*a^2*A - 5*A*b^2 + 2*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*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",11,11,33,0.3333,1,"{3000, 3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
340,1,532,0,1.9342233,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{b \left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \sin (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \left(27 a^2 A b-12 a^3 B+20 a b^2 B-35 A b^3\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(27 a^2 A b-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(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \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-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 \left(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \sin (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{b \left(27 a^2 A b-12 a^3 B+20 a b^2 B-35 A b^3\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(27 a^2 A b-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(-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right) \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-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,"((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*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)]) - ((27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*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]]) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B)*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*(27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*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",12,10,33,0.3030,1,"{3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
341,1,58,0,0.0447992,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 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 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)}}",1,"(2*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",3,3,28,0.1071,1,"{21, 2663, 2661}"
342,1,59,0,0.1462011,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),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 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,"(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",3,3,34,0.08824,1,"{21, 2807, 2805}"
343,1,108,0,0.0819928,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 B \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 b B \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","\frac{2 B \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 b B \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}",1,"(2*B*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*b*B*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,4,28,0.1429,1,"{21, 2664, 2655, 2653}"
344,1,179,0,0.4235763,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b^2 B \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b B \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^2 B \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b B \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*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^2*B*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,34,0.2353,1,"{21, 2802, 3059, 2655, 2653, 12, 2807, 2805}"
345,1,170,0,0.2073669,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{10 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a A+7 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a A+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{10 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a A+7 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a A+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*(9*a*A + 7*b*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*A + 7*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(A*b + a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,6,31,0.1935,1,"{2968, 3023, 2748, 2635, 2639, 2641}"
346,1,140,0,0.1839149,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{2 (7 a A+5 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a A+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 (7 a A+5 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a A+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(6*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*a*A + 5*b*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,6,31,0.1935,1,"{2968, 3023, 2748, 2635, 2641, 2639}"
347,1,108,0,0.1683306,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a A+3 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a A+3 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(5*a*A + 3*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,31,0.1935,1,"{2968, 3023, 2748, 2639, 2635, 2641}"
348,1,75,0,0.1463049,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 (3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 (3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a*A + b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{2968, 3023, 2748, 2641, 2639}"
349,1,71,0,0.1539488,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a A-b B) 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 B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a A-b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*(a*A - b*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,31,0.1613,1,"{2968, 3021, 2748, 2641, 2639}"
350,1,103,0,0.1699011,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 (a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a 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 (a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a 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)*EllipticE[(c + d*x)/2, 2])/d + (2*(a*A + 3*b*B)*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",6,6,31,0.1935,1,"{2968, 3021, 2748, 2636, 2639, 2641}"
351,1,140,0,0.1884759,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a A+5 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a A+5 b B) \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 B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a A+5 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a A+5 b B) \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)}",1,"(-2*(3*a*A + 5*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*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)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,31,0.1935,1,"{2968, 3021, 2748, 2636, 2641, 2639}"
352,1,264,0,0.3812742,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{2 \left(9 a^2 A+14 a b B+7 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 A+14 a b B+7 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \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 A+14 a b B+7 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 A+14 a b B+7 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}",1,"(2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*EllipticF[(c + d*x)/2, 2])/(231*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*B + 11*a*(2*A*b + a*B))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*A*b + 13*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)","A",8,6,33,0.1818,1,"{2990, 3023, 2748, 2635, 2639, 2641}"
353,1,223,0,0.330271,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{2 \left(7 a^2 A+10 a b B+5 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \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 A+10 a b B+5 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}",1,"(2*(7*b^2*B + 9*a*(2*A*b + a*B))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*b^2*B + 9*a*(2*A*b + a*B))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*A*b + 11*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)","A",7,6,33,0.1818,1,"{2990, 3023, 2748, 2635, 2641, 2639}"
354,1,182,0,0.3158701,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \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 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",6,6,33,0.1818,1,"{2990, 3023, 2748, 2639, 2635, 2641}"
355,1,140,0,0.2664903,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a B+2 A b)+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a B+5 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a B+2 A b)+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a B+5 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}",1,"(2*(3*b^2*B + 5*a*(2*A*b + a*B))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*(5*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",5,5,33,0.1515,1,"{2990, 3023, 2748, 2641, 2639}"
356,1,121,0,0.2459156,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-2*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,33,0.1515,1,"{2988, 3023, 2748, 2641, 2639}"
357,1,126,0,0.3050408,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,33,0.1515,1,"{2988, 3021, 2748, 2641, 2639}"
358,1,172,0,0.3505339,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2988, 3021, 2748, 2636, 2639, 2641}"
359,1,305,0,0.5446003,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(26 a^2 B+33 a A b+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}","\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(26 a^2 B+33 a A b+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \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*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b^2*(11*A*b + 15*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)","A",8,7,33,0.2121,1,"{2990, 3033, 3023, 2748, 2635, 2641, 2639}"
360,1,255,0,0.4964177,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \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 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \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*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*A*b + 13*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)","A",7,7,33,0.2121,1,"{2990, 3033, 3023, 2748, 2639, 2635, 2641}"
361,1,205,0,0.4777619,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}","\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)","A",6,6,33,0.1818,1,"{2990, 3033, 3023, 2748, 2641, 2639}"
362,1,202,0,0.4637677,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 A-3 a b B-A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}-\frac{2 b^2 (5 a A-b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 A-3 a b B-A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}-\frac{2 b^2 (5 a A-b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}",1,"(-2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a^2*A - A*b^2 - 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*a*A - b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{2989, 3033, 3023, 2748, 2641, 2639}"
363,1,192,0,0.4654979,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (3 a B+7 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (3 a B+7 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \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*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(a*A - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,33,0.1818,1,"{2989, 3031, 3023, 2748, 2641, 2639}"
364,1,204,0,0.4815204,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(3 a^2 A+15 a b B+14 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \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 A b+a^3 B+9 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(3 a^2 A+15 a b B+14 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \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*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(9*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*a^2*A + 14*A*b^2 + 15*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,6,33,0.1818,1,"{2989, 3031, 3021, 2748, 2641, 2639}"
365,1,182,0,0.819539,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 \left(3 a^2+b^2\right) (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a^3 (A b-a B) \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 b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}","\frac{2 \left(3 a^2+b^2\right) (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a^3 (A b-a B) \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 b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"(-2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) + (2*(3*a^2 + b^2)*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(3*b^4*d) - (2*a^3*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^4*(a + b)*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,33,0.2121,1,"{2990, 3049, 3059, 2639, 3002, 2641, 2805}"
366,1,137,0,0.513183,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","-\frac{2 \left(-3 a^2 B+3 a A b-b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (A b-a B) \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 b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","-\frac{2 \left(-3 a^2 B+3 a A b-b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (A b-a B) \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 b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) + (2*a^2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,33,0.1818,1,"{2990, 3059, 2639, 3002, 2641, 2805}"
367,1,89,0,0.2096881,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(b^2*d) - (2*a*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",5,5,33,0.1515,1,"{3002, 2639, 2803, 2641, 2805}"
368,1,61,0,0.1437134,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*B*EllipticF[(c + d*x)/2, 2])/(b*d) + (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)","A",3,3,33,0.09091,1,"{3002, 2641, 2805}"
369,1,86,0,0.3091942,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a 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 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a 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)}}",1,"(-2*A*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])","A",5,5,33,0.1515,1,"{3000, 3059, 2639, 12, 2805}"
370,1,150,0,0.7700018,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{2 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (A b-a B) \Pi \left(\frac{2 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)}{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 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (A b-a B) \Pi \left(\frac{2 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)}{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*b*(A*b - a*B)*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",7,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
371,1,303,0,0.9330817,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","-\frac{\left(9 a^3 A b+16 a^2 b^2 B-15 a^4 B-12 a A b^3+2 b^4 B\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 A b-5 a^3 B+4 a b^2 B-2 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 A b-5 a^3 B+7 a b^2 B-5 A b^3\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 (A b-a B) \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 B+3 a A b+2 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}","-\frac{\left(9 a^3 A b+16 a^2 b^2 B-15 a^4 B-12 a A b^3+2 b^4 B\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 A b-5 a^3 B+4 a b^2 B-2 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 A b-5 a^3 B+7 a b^2 B-5 A b^3\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 (A b-a B) \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 B+3 a A b+2 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}",1,"((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2989, 3049, 3059, 2639, 3002, 2641, 2805}"
372,1,224,0,0.6270272,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 A b-3 a^3 B+4 a b^2 B-2 A b^3\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 B+a A b+2 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 A b-3 a^3 B+5 a b^2 B-3 A b^3\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 (A b-a B) \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 A b-3 a^3 B+4 a b^2 B-2 A b^3\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 B+a A b+2 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 A b-3 a^3 B+5 a b^2 B-3 A b^3\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 (A b-a B) \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*A*b - 3*a^2*B + 2*b^2*B)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{2989, 3059, 2639, 3002, 2641, 2805}"
373,1,198,0,0.5404712,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2 B+a A b-2 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 A b+a^3 B-3 a b^2 B+A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}-\frac{(A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{\left(a^2 B+a A b-2 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 A b+a^3 B-3 a b^2 B+A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}-\frac{(A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{2999, 3059, 2639, 3002, 2641, 2805}"
374,1,200,0,0.6181383,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","-\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b+a^3 (-B)-a b^2 B-A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b+a^3 (-B)-a b^2 B-A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-(((A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,33,0.1818,1,"{3000, 3059, 2639, 3002, 2641, 2805}"
375,1,256,0,0.9229181,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 A+a b B-3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 A b-3 a^3 B+a b^2 B-3 A b^3\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 A+a b B-3 A b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}","\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 A+a b B-3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 A b-3 a^3 B+a b^2 B-3 A b^3\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 A+a b B-3 A b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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*A - 3*A*b^2 + a*b*B)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
376,1,345,0,1.2921303,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\left(2 a^2 A+3 a b B-5 A b^2\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 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\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 (A b-a B) \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 A+3 a b B-5 A b^2\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 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","\frac{\left(2 a^2 A+3 a b B-5 A b^2\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 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\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 (A b-a B) \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 A+3 a b B-5 A b^2\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 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",8,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
377,1,367,0,1.0128594,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-5 a^2 A b^3+3 a^4 A b+33 a^3 b^2 B-15 a^5 B-24 a b^4 B+8 A b^5\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(3 a^3 A b+29 a^2 b^2 B-15 a^4 B-9 a A b^3-8 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-35 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 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 (A b-a B) \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 A b-5 a^3 B+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}","\frac{\left(-5 a^2 A b^3+3 a^4 A b+33 a^3 b^2 B-15 a^5 B-24 a b^4 B+8 A b^5\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(3 a^3 A b+29 a^2 b^2 B-15 a^4 B-9 a A b^3-8 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-35 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 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 (A b-a B) \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 A b-5 a^3 B+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"-((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*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*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*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",7,7,33,0.2121,1,"{2989, 3047, 3059, 2639, 3002, 2641, 2805}"
378,1,344,0,0.9896753,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^3 A b-5 a^2 b^2 B+3 a^4 B-7 a A b^3+8 b^4 B\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 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right) \Pi \left(\frac{2 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 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","\frac{\left(a^3 A b-5 a^2 b^2 B+3 a^4 B-7 a A b^3+8 b^4 B\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 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right) \Pi \left(\frac{2 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 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2989, 3055, 3059, 2639, 3002, 2641, 2805}"
379,1,337,0,0.919423,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\left(3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2 A b+a^3 (-B)-5 a b^2 B+A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-3 a b^4 B-A b^5\right) \Pi \left(\frac{2 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 A b+a^3 (-B)-5 a b^2 B+A b^3\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{(A b-a B) \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 A b+a^3 B-7 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2 A b+a^3 (-B)-5 a b^2 B+A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-3 a b^4 B-A b^5\right) \Pi \left(\frac{2 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 A b+a^3 (-B)-5 a b^2 B+A b^3\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{(A b-a B) \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*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((A*b - a*B)*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*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{2999, 3055, 3059, 2639, 3002, 2641, 2805}"
380,1,345,0,1.0599978,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","-\frac{\left(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+a b^4 B+3 A b^5\right) \Pi \left(\frac{2 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 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+a b^4 B+3 A b^5\right) \Pi \left(\frac{2 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 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
381,1,420,0,1.4738486,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 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 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+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{b \left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\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 (A b-a B) \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 A b-7 a^3 B+a b^2 B-5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 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 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+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{b \left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\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 (A b-a B) \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*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*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*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*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",8,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
382,1,523,0,1.9525551,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b \left(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-15 a b^4 B+35 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b \left(13 a^2 A b-9 a^3 B+3 a b^2 B-7 A b^3\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{b (A b-a B) \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{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}","\frac{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b \left(-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-15 a b^4 B+35 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b \left(13 a^2 A b-9 a^3 B+3 a b^2 B-7 A b^3\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{b (A b-a B) \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{\left(-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF[(c + d*x)/2, 2])/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",9,7,33,0.2121,1,"{3000, 3055, 3059, 2639, 3002, 2641, 2805}"
383,1,44,0,0.0248216,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),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{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",3,3,36,0.08333,1,"{21, 2635, 2639}"
384,1,44,0,0.0236041,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",3,3,36,0.08333,1,"{21, 2635, 2641}"
385,1,17,0,0.0109797,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 B E\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}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d","A",2,2,36,0.05556,1,"{21, 2639}"
386,1,17,0,0.0112141,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 B F\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}",1,"(2*B*EllipticF[(c + d*x)/2, 2])/d","A",2,2,36,0.05556,1,"{21, 2641}"
387,1,40,0,0.0216924,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","\frac{2 B \sin (c+d x)}{d \sqrt{\cos (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 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",3,3,36,0.08333,1,"{21, 2636, 2639}"
388,1,44,0,0.0229834,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),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 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*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",3,3,36,0.08333,1,"{21, 2636, 2641}"
389,1,116,0,0.4028926,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{2 B \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a^3 B \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 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 B \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a^3 B \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 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(-2*a*B*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*(3*a^2 + b^2)*B*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) - (2*a^3*B*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",7,7,36,0.1944,1,"{21, 2793, 3059, 2639, 3002, 2641, 2805}"
390,1,78,0,0.166554,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{2 a^2 B \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 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 a^2 B \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 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*B*EllipticF[(c + d*x)/2, 2])/(b^2*d) + (2*a^2*B*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",6,6,36,0.1667,1,"{21, 2804, 2639, 2803, 2641, 2805}"
391,1,55,0,0.1029772,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}",1,"(2*B*EllipticF[(c + d*x)/2, 2])/(b*d) - (2*a*B*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)","A",4,4,36,0.1111,1,"{21, 2803, 2641, 2805}"
392,1,30,0,0.0487697,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{2 B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*B*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",2,2,36,0.05556,1,"{21, 2805}"
393,1,80,0,0.2467791,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","-\frac{2 b B \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 \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*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",6,6,36,0.1667,1,"{21, 2802, 3059, 2639, 12, 2805}"
394,1,133,0,0.5597793,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{2 b^2 B \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b B \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^2 B \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b B \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*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*b^2*B*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*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])","A",8,8,36,0.2222,1,"{21, 2802, 3055, 3059, 2639, 3002, 2641, 2805}"
395,1,560,0,1.5073444,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\left(-3 a^2 B+6 a A b+16 b^2 B\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 B+6 a A b+16 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)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(2 a^2 A b+a^3 (-B)-4 a b^2 B-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}} \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 B+6 A b+8 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)}{24 b^2 d}+\frac{(2 A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{B \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 B+6 a A b+16 b^2 B\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 B+6 a A b+16 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)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(2 a^2 A b+a^3 (-B)-4 a b^2 B-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}} \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 B+6 A b+8 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)}{24 b^2 d}+\frac{(2 A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{B \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*A*b - 3*a^2*B + 16*b^2*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)])/(24*a*b^2*d) + (Sqrt[a + b]*(a + 2*b)*(6*A*b - 3*a*B + 8*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)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*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)])/(8*b^3*d) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*A*b - a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)","A",8,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
396,1,473,0,1.0415178,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(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",7,7,35,0.2000,1,"{3003, 3061, 3053, 2809, 2998, 2816, 2994}"
397,1,385,0,0.7132927,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a+b} (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)}{d}-\frac{\sqrt{a+b} (a B+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}} \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{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{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{\sqrt{a+b} (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)}{d}-\frac{\sqrt{a+b} (a B+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}} \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{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{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}",1,"-(((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)) + (Sqrt[a + b]*(2*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 - (Sqrt[a + b]*(2*A*b + a*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)])/(b*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,35,0.1714,1,"{3003, 3053, 2809, 2998, 2816, 2994}"
398,1,351,0,0.5041363,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \sqrt{a+b} (A b-a (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 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 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}} \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} (A b-a (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 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 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}} \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)*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) + (2*Sqrt[a + b]*(A*b - a*(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]*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)])/d","A",5,5,35,0.1429,1,"{2991, 2809, 2998, 2816, 2994}"
399,1,284,0,0.5052396,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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 (a-b) \sqrt{a+b} (A-3 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 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 (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 (a-b) \sqrt{a+b} (A-3 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 d}+\frac{2 A \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]*(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*(a - b)*Sqrt[a + b]*(A - 3*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)])/(3*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,35,0.1143,1,"{2999, 2998, 2816, 2994}"
400,1,350,0,0.8249122,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+5 a b 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}} E\left(\sin ^{-1}\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-5 a B+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 (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} \left(9 a^2 A+5 a b 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}} E\left(\sin ^{-1}\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-5 a B+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 (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]*(9*a^2*A - 2*A*b^2 + 5*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)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b - 5*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)])/(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",5,5,35,0.1429,1,"{2999, 3055, 2998, 2816, 2994}"
401,1,433,0,1.1756395,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+7 a b B-4 A b^2\right) \sin (c+d x) \sqrt{a+b \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 A-63 B)+2 a b (3 A-7 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 (a-b) \sqrt{a+b} \left(19 a^2 A b+63 a^3 B-14 a b^2 B+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}} E\left(\sin ^{-1}\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 \left(25 a^2 A+7 a b B-4 A b^2\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 A-63 B)+2 a b (3 A-7 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 (a-b) \sqrt{a+b} \left(19 a^2 A b+63 a^3 B-14 a b^2 B+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}} E\left(\sin ^{-1}\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]*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + a^2*(25*A - 63*B) + 2*a*b*(3*A - 7*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)])/(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*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))","A",6,5,35,0.1429,1,"{2999, 3055, 2998, 2816, 2994}"
402,1,670,0,2.0960994,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{\left(-3 a^2 B+8 a A b+12 b^2 B\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 A b-9 a^3 B+156 a b^2 B+128 A b^3\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 A+B)+9 a^3 B-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 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)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 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}} E\left(\sin ^{-1}\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(8 a^3 A b-24 a^2 b^2 B-3 a^4 B-96 a A b^3-48 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}} \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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{B \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 B+8 a A b+12 b^2 B\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 A b-9 a^3 B+156 a b^2 B+128 A b^3\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 A+B)+9 a^3 B-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 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)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 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}} E\left(\sin ^{-1}\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(8 a^3 A b-24 a^2 b^2 B-3 a^4 B-96 a A b^3-48 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}} \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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{B \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*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*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)])/(192*a*b^2*d) - (Sqrt[a + b]*(9*a^3*B - 6*a^2*b*(4*A + B) - 8*b^3*(16*A + 9*B) - 4*a*b^2*(28*A + 39*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)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*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)])/(64*b^3*d) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((8*a*A*b - 3*a^2*B + 12*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)","A",9,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
403,1,566,0,1.6631129,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\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 B+30 a A b+14 a b B+12 A b^2+16 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)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 B+30 a A b+16 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)}{24 a b d}-\frac{\sqrt{a+b} \left(6 a^2 A b+a^3 (-B)+12 a b^2 B+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}} \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 B+6 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b B \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 B+30 a A b+16 b^2 B\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 B+30 a A b+14 a b B+12 A b^2+16 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)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 B+30 a A b+16 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)}{24 a b d}-\frac{\sqrt{a+b} \left(6 a^2 A b+a^3 (-B)+12 a b^2 B+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}} \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 B+6 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b B \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*A*b + 3*a^2*B + 16*b^2*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)])/(24*a*b*d) + (Sqrt[a + b]*(30*a*A*b + 12*A*b^2 + 3*a^2*B + 14*a*b*B + 16*b^2*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)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*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)])/(8*b^2*d) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((6*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*B*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
404,1,472,0,1.1469596,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","-\frac{\sqrt{a+b} \left(3 a^2 B+12 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 d}+\frac{(5 a B+4 A 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 A+5 a B+4 A b+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)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 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 d}+\frac{b B \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 B+12 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 d}+\frac{(5 a B+4 A 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 A+5 a B+4 A b+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)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 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 d}+\frac{b 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 + 5*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*d) + (Sqrt[a + b]*(8*a*A + 4*A*b + 5*a*B + 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*d) - (Sqrt[a + b]*(12*a*A*b + 3*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*d) + ((4*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,35,0.2000,1,"{2990, 3061, 3053, 2809, 2998, 2816, 2994}"
405,1,449,0,1.1758113,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","-\frac{(2 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (A-B)-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)}{d}+\frac{(a-b) \sqrt{a+b} (2 a A-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)}{a d}-\frac{\sqrt{a+b} (3 a B+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}} \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 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","-\frac{(2 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (A-B)-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)}{d}+\frac{(a-b) \sqrt{a+b} (2 a A-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)}{a d}-\frac{\sqrt{a+b} (3 a B+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}} \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 A \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*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) - (Sqrt[a + b]*(2*a*(A - B) - b*(4*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 - (Sqrt[a + b]*(2*A*b + 3*a*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)])/d + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,35,0.2000,1,"{2989, 3061, 3053, 2809, 2998, 2816, 2994}"
406,1,419,0,0.8583423,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b} \left(a^2 (A-3 B)-a (4 A b-6 b B)+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}} F\left(\sin ^{-1}\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 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)}{3 a d}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b 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}} \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 (A-3 B)-a (4 A b-6 b B)+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}} F\left(\sin ^{-1}\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 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)}{3 a d}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b 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}} \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*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*d) + (2*Sqrt[a + b]*(3*A*b^2 + a^2*(A - 3*B) - a*(4*A*b - 6*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)])/(3*a*d) - (2*b*Sqrt[a + b]*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)])/d + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,35,0.1714,1,"{2989, 3053, 2809, 2998, 2816, 2994}"
407,1,353,0,0.9243908,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+20 a b B+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)}{15 a^2 d}+\frac{2 (5 a B+6 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-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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 d}+\frac{2 a 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(9 a^2 A+20 a b B+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)}{15 a^2 d}+\frac{2 (5 a B+6 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-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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 d}+\frac{2 a 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]*(9*a^2*A + 3*A*b^2 + 20*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)])/(15*a^2*d) - (2*(a - b)*Sqrt[a + b]*(9*a*A - 3*A*b - 5*a*B + 15*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)])/(15*a*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",5,5,35,0.1429,1,"{2989, 3055, 2998, 2816, 2994}"
408,1,433,0,1.3461638,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+42 a b B+3 A b^2\right) \sin (c+d x) \sqrt{a+b \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 A-63 B))+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(82 a^2 A b+63 a^3 B+21 a b^2 B-6 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}} E\left(\sin ^{-1}\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+8 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 A \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 A+42 a b B+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(a^2 (-(25 A-63 B))+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(82 a^2 A b+63 a^3 B+21 a b^2 B-6 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}} E\left(\sin ^{-1}\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+8 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 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]*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B) + 3*a*b*(19*A - 7*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)])/(105*a^2*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(8*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*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))","A",6,5,35,0.1429,1,"{2989, 3055, 2998, 2816, 2994}"
409,1,522,0,1.8845842,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(88 a^2 A b+75 a^3 B+9 a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2 A+72 a b B+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b (13 A-57 B)+a^3 (-(147 A-75 B))+6 a b^2 (A-3 B)+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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \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 (9 a B+10 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \left(88 a^2 A b+75 a^3 B+9 a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2 A+72 a b B+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 b (13 A-57 B)+a^3 (-(147 A-75 B))+6 a b^2 (A-3 B)+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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \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 (9 a B+10 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*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 - a^3*(147*A - 75*B) + 3*a^2*b*(13*A - 57*B) + 6*a*b^2*(A - 3*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)])/(315*a^3*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2))","A",7,5,35,0.1429,1,"{2989, 3055, 2998, 2816, 2994}"
410,1,779,0,3.0825054,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{\left(-15 a^2 B+50 a A b+64 b^2 B\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 A b-15 a^3 B+172 a b^2 B+120 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 b^4 B\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 A+423 B)-30 a^3 b (5 A+B)+45 a^4 B-8 a b^3 (355 A+193 B)-16 b^4 (45 A+64 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)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 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)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-240 a^2 A b^3+10 a^4 A b-40 a^3 b^2 B-3 a^5 B-240 a b^4 B-96 A b^5\right) \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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{B \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 B+50 a A b+64 b^2 B\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 A b-15 a^3 B+172 a b^2 B+120 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 b^4 B\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 A+423 B)-30 a^3 b (5 A+B)+45 a^4 B-8 a b^3 (355 A+193 B)-16 b^4 (45 A+64 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)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 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)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-240 a^2 A b^3+10 a^4 A b-40 a^3 b^2 B-3 a^5 B-240 a b^4 B-96 A b^5\right) \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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{B \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*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*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)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*B - 30*a^3*b*(5*A + B) - 16*b^4*(45*A + 64*B) - 8*a*b^3*(355*A + 193*B) - 4*a^2*b^2*(295*A + 423*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)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*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)])/(128*b^3*d) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*A*b + 120*A*b^3 - 15*a^3*B + 172*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((50*a*A*b - 15*a^2*B + 64*b^2*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)","A",10,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
411,1,664,0,2.2058651,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{\left(5 a^2 B+24 a A b+12 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (132 A+59 B)+15 a^3 B+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 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)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 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}} E\left(\sin ^{-1}\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(40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right) \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 B+8 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b B \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 B+24 a A b+12 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(2 a^2 b (132 A+59 B)+15 a^3 B+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 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)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 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}} E\left(\sin ^{-1}\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(40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right) \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 B+8 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b B \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*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*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)])/(192*a*b*d) + (Sqrt[a + b]*(15*a^3*B + 8*b^3*(16*A + 9*B) + 2*a^2*b*(132*A + 59*B) + 4*a*b^2*(52*A + 71*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)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*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)])/(64*b^2*d) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((24*a*A*b + 5*a^2*B + 12*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",9,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
412,1,564,0,1.6984043,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (48 A+33 B)+a (54 A b+26 b B)+4 b^2 (3 A+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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 B+54 a A b+16 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)}{24 a d}-\frac{\sqrt{a+b} \left(30 a^2 A b+5 a^3 B+20 a b^2 B+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}} \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 B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (48 A+33 B)+a (54 A b+26 b B)+4 b^2 (3 A+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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 B+54 a A b+16 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)}{24 a d}-\frac{\sqrt{a+b} \left(30 a^2 A b+5 a^3 B+20 a b^2 B+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}} \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 B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b B \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*A*b + 33*a^2*B + 16*b^2*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)])/(24*a*d) + (Sqrt[a + b]*(4*b^2*(3*A + 4*B) + a^2*(48*A + 33*B) + a*(54*A*b + 26*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)])/(24*d) - (Sqrt[a + b]*(30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*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)])/(8*b*d) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,35,0.2286,1,"{2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
413,1,547,0,1.665114,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","-\frac{\left(8 a^2 A-9 a b B-4 A b^2\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 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+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)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 B+20 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 d}-\frac{b (4 a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}","-\frac{\left(8 a^2 A-9 a b B-4 A b^2\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 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+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)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 B+20 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 d}-\frac{b (4 a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a 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^2*A - 4*A*b^2 - 9*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)])/(4*a*d) - (Sqrt[a + b]*(8*a^2*(A - B) - 2*b^2*(2*A + B) - 3*a*b*(8*A + 3*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*d) - (Sqrt[a + b]*(20*a*A*b + 15*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*d) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*a*A - b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,8,35,0.2286,1,"{2989, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
414,1,536,0,1.6688658,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","-\frac{\left(6 a^2 B+14 a A b-3 b^2 B\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 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+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 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-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)}{3 a d}+\frac{2 a (a B+2 A 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 B+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}} \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 A \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 B+14 a A b-3 b^2 B\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 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+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 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-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)}{3 a d}+\frac{2 a (a B+2 A 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 B+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}} \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 A \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*A*b + 6*a^2*B - 3*b^2*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*d) - (Sqrt[a + b]*(2*a*b*(7*A - 9*B) - 2*a^2*(A - 3*B) - 3*b^2*(6*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)])/(3*d) - (b*Sqrt[a + b]*(2*A*b + 5*a*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)])/d + (2*a*(2*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,8,35,0.2286,1,"{2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
415,1,493,0,1.2466387,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 \sqrt{a+b} \left(a^2 b (17 A-35 B)+a^3 (-(9 A-5 B))-a b^2 (23 A-45 B)+15 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{2 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d}+\frac{2 a (5 a B+8 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 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^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}} \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 (17 A-35 B)+a^3 (-(9 A-5 B))-a b^2 (23 A-45 B)+15 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{2 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d}+\frac{2 a (5 a B+8 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 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^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}} \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]*(9*a^2*A + 23*A*b^2 + 35*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)])/(15*a*d) + (2*Sqrt[a + b]*(15*A*b^3 - a*b^2*(23*A - 45*B) + a^2*b*(17*A - 35*B) - a^3*(9*A - 5*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)])/(15*a*d) - (2*b^2*Sqrt[a + b]*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)])/d + (2*a*(8*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*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,35,0.2000,1,"{2989, 3047, 3053, 2809, 2998, 2816, 2994}"
416,1,434,0,1.3718482,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 A b+63 a^3 B+161 a b^2 B+15 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}} E\left(\sin ^{-1}\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 B+10 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 A \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 A+77 a b B+45 A b^2\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 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 A b+63 a^3 B+161 a b^2 B+15 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}} E\left(\sin ^{-1}\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 B+10 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 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]*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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)])/(105*a^2*d) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*A - 63*B) + 15*b^2*(A - 7*B) - 8*a*b*(15*A - 7*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)])/(105*a*d) + (2*a*(10*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*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",6,6,35,0.1714,1,"{2989, 3047, 3055, 2998, 2816, 2994}"
417,1,522,0,1.9388852,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(163 a^2 A b+75 a^3 B+135 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2 A+135 a b B+75 A b^2\right) \sin (c+d x) \sqrt{a+b \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-60 B)+3 a^3 (49 A-25 B)+15 a b^2 (11 A-3 B)+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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \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 B+4 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 A \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 A b+75 a^3 B+135 a b^2 B+5 A b^3\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 A+135 a b B+75 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-60 B)+3 a^3 (49 A-25 B)+15 a b^2 (11 A-3 B)+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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \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 B+4 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 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]*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 - 6*a^2*b*(19*A - 60*B) + 3*a^3*(49*A - 25*B) + 15*a*b^2*(11*A - 3*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)])/(315*a^2*d) + (2*a*(4*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*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",7,6,35,0.1714,1,"{2989, 3047, 3055, 2998, 2816, 2994}"
418,1,622,0,2.6247963,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(13/2),x]","\frac{2 \left(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sqrt{a+b \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 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \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 B+14 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \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 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sqrt{a+b \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 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \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 B+14 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \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*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*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*A*b^4 + 3*a^4*(225*A - 539*B) - 6*a^3*b*(505*A - 209*B) + 15*a^2*b^2*(19*A - 121*B) + 10*a*b^3*(3*A - 11*B))*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*A*b + 11*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a*d*Cos[c + d*x]^(5/2)) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))","A",8,6,35,0.1714,1,"{2989, 3047, 3055, 2998, 2816, 2994}"
419,1,418,0,0.9495077,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(\frac{3 b B}{2 a}+B \cos (c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","-\frac{B (a-3 b) \sqrt{a+b} \left(2 a^2-a b+3 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 d}+\frac{2 B (a-b) \sqrt{a+b} \left(a^2+3 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 d}-\frac{b B \sqrt{a+b} \left(\frac{3 b^2}{a}+5 a\right) \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{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{B (a-3 b) \sqrt{a+b} \left(2 a^2-a b+3 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 d}+\frac{2 B (a-b) \sqrt{a+b} \left(a^2+3 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 d}-\frac{b B \sqrt{a+b} \left(\frac{3 b^2}{a}+5 a\right) \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{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(a^2 + 3*b^2)*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) - ((a - 3*b)*Sqrt[a + b]*(2*a^2 - a*b + 3*b^2)*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) - (b*Sqrt[a + b]*(5*a + (3*b^2)/a)*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)])/d + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",6,6,43,0.1395,1,"{2989, 2991, 2809, 2998, 2816, 2994}"
420,1,479,0,1.0763634,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} \left(-3 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^3 d}+\frac{(4 A b-3 a B) \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 B+4 A b+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)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 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)}{4 a b^2 d}+\frac{B \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 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^3 d}+\frac{(4 A b-3 a B) \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 B+4 A b+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)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 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)}{4 a b^2 d}+\frac{B \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*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)])/(4*a*b^2*d) + (Sqrt[a + b]*(4*A*b - 3*a*B + 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^2*d) + (Sqrt[a + b]*(4*a*A*b - 3*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^3*d) + ((4*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",7,7,35,0.2000,1,"{2990, 3061, 3053, 2809, 2998, 2816, 2994}"
421,1,427,0,1.0882572,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{\sqrt{a+b} (2 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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}","-\frac{\sqrt{a+b} (2 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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}",1,"-(((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*b*d)) + (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)])/(b*d) - (Sqrt[a + b]*(2*A*b - a*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)])/(b^2*d) + (a*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,35,0.2000,1,"{3003, 3051, 2809, 2993, 2998, 2816, 2994}"
422,1,228,0,0.273712,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\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 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}} \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 \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 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}} \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*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]*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)])/(b*d)","A",3,3,35,0.08571,1,"{3006, 2809, 2816}"
423,1,230,0,0.3159601,"\int \frac{A+B \cos (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])/(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 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}",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)","A",3,3,35,0.08571,1,"{2998, 2816, 2994}"
424,1,290,0,0.5235995,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} (a (A-3 B)+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{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} (a (A-3 B)+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{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))*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",4,4,35,0.1143,1,"{3000, 2998, 2816, 2994}"
425,1,363,0,0.8621244,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","-\frac{2 \sqrt{a+b} \left(a^2 (9 A-5 B)-2 a b (A+5 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(9 a^2 A-10 a b 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}} E\left(\sin ^{-1}\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} \left(a^2 (9 A-5 B)-2 a b (A+5 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(9 a^2 A-10 a b 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}} E\left(\sin ^{-1}\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]*(9*a^2*A + 8*A*b^2 - 10*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)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*A*b^2 + a^2*(9*A - 5*B) - 2*a*b*(A + 5*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)])/(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",5,5,35,0.1429,1,"{3000, 3055, 2998, 2816, 2994}"
426,1,500,0,1.286972,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{\left(-3 a^2 B+2 a A b+b^2 B\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 (A b-a B) \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 B+2 a A b+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 b^2 d \sqrt{a+b}}-\frac{(2 A b-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}} F\left(\sin ^{-1}\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 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}} \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 B+2 a A b+b^2 B\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 (A b-a B) \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 B+2 a A b+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 b^2 d \sqrt{a+b}}-\frac{(2 A b-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}} F\left(\sin ^{-1}\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 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}} \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*A*b - 3*a^2*B + b^2*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*b^2*Sqrt[a + b]*d) - ((2*A*b - (3*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)])/(b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*A*b - 3*a*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)])/(b^3*d) + (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",7,7,35,0.2000,1,"{2989, 3061, 3053, 2809, 2998, 2816, 2994}"
427,1,416,0,0.6118898,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (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 b d \sqrt{a+b}}-\frac{2 (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)}{a b d \sqrt{a+b}}-\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}} \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 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (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 b d \sqrt{a+b}}-\frac{2 (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)}{a b d \sqrt{a+b}}-\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}} \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 - 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*b*Sqrt[a + b]*d) + (2*(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*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*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)])/(b^2*d) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{2992, 2809, 2794, 2795, 2816, 2994}"
428,1,284,0,0.5116177,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{a^2 d \sqrt{a+b}}+\frac{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 \sqrt{a+b}}","-\frac{2 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{a^2 d \sqrt{a+b}}+\frac{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 \sqrt{a+b}}",1,"(2*(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)])/(a^2*Sqrt[a + b]*d) + (2*(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*Sqrt[a + b]*d) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{2993, 2998, 2816, 2994}"
429,1,305,0,0.6145303,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 A+a b 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}} E\left(\sin ^{-1}\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-B)+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 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 A+a b 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}} E\left(\sin ^{-1}\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-B)+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*(a^2*A - 2*A*b^2 + 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^3*Sqrt[a + b]*d) - (2*(2*A*b + a*(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^2*Sqrt[a + b]*d) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{3000, 2998, 2816, 2994}"
430,1,393,0,0.9801146,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \left(a^2 A+3 a b B-4 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(5 a^2 A b-3 a^3 B+6 a b^2 B-8 A b^3\right) \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 (A-3 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)}{3 a^3 d \sqrt{a+b}}","\frac{2 \left(a^2 A+3 a b B-4 A b^2\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 (A b-a B) \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 A b-3 a^3 B+6 a b^2 B-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}} E\left(\sin ^{-1}\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 (A-3 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)}{3 a^3 d \sqrt{a+b}}",1,"(-2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*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^4*Sqrt[a + b]*d) + (2*(a + 2*b)*(4*A*b + a*(A - 3*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)])/(3*a^3*Sqrt[a + b]*d) + (2*b*(A*b - a*B)*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*A - 4*A*b^2 + 3*a*b*B)*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",5,5,35,0.1429,1,"{3000, 3055, 2998, 2816, 2994}"
431,1,674,0,2.1890305,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 a (A b-a B) \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 A b-5 a^3 B+9 a b^2 B-6 A b^3\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(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(6 a^2 A b-5 a^2 b B-15 a^3 B+2 a A b^2+21 a b^2 B-12 A b^3+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 b^3 d (a-b) (a+b)^{3/2}}+\frac{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \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 A b-5 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 a (A b-a B) \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 A b-5 a^3 B+9 a b^2 B-6 A b^3\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(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(6 a^2 A b-5 a^2 b B-15 a^3 B+2 a A b^2+21 a b^2 B-12 A b^3+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 b^3 d (a-b) (a+b)^{3/2}}+\frac{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \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 A b-5 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,"((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*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*(a - b)*b^3*(a + b)^(3/2)*d) - ((6*a^2*A*b + 2*a*A*b^2 - 12*A*b^3 - 15*a^3*B - 5*a^2*b*B + 21*a*b^2*B + 3*b^3*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)])/(3*(a - b)*b^3*(a + b)^(3/2)*d) - (Sqrt[a + b]*(2*A*b - 5*a*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)])/(b^4*d) + (2*a*(A*b - a*B)*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*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*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*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*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",8,8,35,0.2286,1,"{2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
432,1,545,0,1.4018908,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 B-3 a^3 B+a A b^2+6 a b^2 B-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)}{3 a b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^3 B-7 a b^2 B+4 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}} E\left(\sin ^{-1}\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 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}} \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 B-7 a b^2 B+4 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 B-3 a^3 B+a A b^2+6 a b^2 B-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)}{3 a b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^3 B-7 a b^2 B+4 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}} E\left(\sin ^{-1}\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 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}} \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*A*b^3 + 3*a^3*B - 7*a*b^2*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*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*A*b^2 - 3*A*b^3 - 3*a^3*B - a^2*b*B + 6*a*b^2*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)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*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)])/(b^3*d) + (2*a*(A*b - a*B)*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*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",7,7,35,0.2000,1,"{2989, 3051, 2809, 2993, 2998, 2816, 2994}"
433,1,391,0,0.8685695,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(3 a^2 A-4 a b B+A b^2\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 (A b-a B) \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 A-4 a b B+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^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a A+a B-A b-3 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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 A-4 a b B+A b^2\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 (A b-a B) \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 A-4 a b B+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^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a A+a B-A b-3 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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*A + A*b^2 - 4*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)])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*(3*a*A - A*b + a*B - 3*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + 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*(A*b - a*B)*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*A + A*b^2 - 4*a*b*B)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2999, 2993, 2998, 2816, 2994}"
434,1,429,0,0.9854628,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","-\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 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}} E\left(\sin ^{-1}\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 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 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}} E\left(\sin ^{-1}\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)*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^2*(A + B) + a*b*(3*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)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*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*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*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",5,5,35,0.1429,1,"{3000, 2993, 2998, 2816, 2994}"
435,1,456,0,1.1609057,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 b \left(8 a^2 A b-5 a^3 B+a b^2 B-4 A b^3\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 A+B)-3 a^3 (A-B)+2 a b^2 (3 A-B)+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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \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 A b-5 a^3 B+a b^2 B-4 A b^3\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 A+B)-3 a^3 (A-B)+2 a b^2 (3 A-B)+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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \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*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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^4*(a - b)*(a + b)^(3/2)*d) + (2*(8*A*b^3 - 3*a^3*(A - B) + 2*a*b^2*(3*A - B) - 3*a^2*b*(3*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)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*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*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{3000, 3055, 2998, 2816, 2994}"
436,1,567,0,1.8768727,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(10 a^2 A b-7 a^3 B+3 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \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+3 B)-9 a^3 b (A-B)+a^4 (-(A-3 B))+4 a b^3 (3 A-2 B)+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{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \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 (a-b) (a+b)^{3/2}}","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(10 a^2 A b-7 a^3 B+3 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \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+3 B)-9 a^3 b (A-B)+a^4 (-(A-3 B))+4 a b^3 (3 A-2 B)+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{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \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 (a-b) (a+b)^{3/2}}",1,"(-2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*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*(a - b)*(a + b)^(3/2)*d) - (2*(16*A*b^4 - a^4*(A - 3*B) + 4*a*b^3*(3*A - 2*B) - 9*a^3*b*(A - B) - 2*a^2*b^2*(8*A + 3*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)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*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*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))","A",6,5,35,0.1429,1,"{3000, 3055, 2998, 2816, 2994}"
437,1,419,0,0.7888814,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}","\frac{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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}",1,"-(((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*b*d)) + (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)])/(b*d) + (a*Sqrt[a + b]*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)])/(b^2*d) + (a*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])","A",9,9,38,0.2368,1,"{21, 2820, 2809, 3003, 2993, 12, 2801, 2816, 2994}"
438,1,117,0,0.0845342,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),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}} \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}} \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]*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",2,2,38,0.05263,1,"{21, 2809}"
439,1,110,0,0.0868031,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),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 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}",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)","A",2,2,38,0.05263,1,"{21, 2816}"
440,1,226,0,0.2683292,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),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 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 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 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}",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*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",4,4,38,0.1053,1,"{21, 2801, 2816, 2994}"
441,1,72,0,0.0876338,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{2+3 \cos (c+d x)}} \, dx","Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]),x]","-\frac{\cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{d}","-\frac{\cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{d}",1,"-((Cot[c + d*x]*EllipticE[ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/d)","A",1,1,33,0.03030,1,"{2994}"
442,1,70,0,0.103997,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-2+3 \cos (c+d x)}} \, dx","Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]]),x]","-\frac{\sqrt{5} \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}","-\frac{\sqrt{5} \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}",1,"-((Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d)","A",1,1,33,0.03030,1,"{2994}"
443,1,93,0,0.2064882,"\int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(1 + Cos[c + d*x])/(Sqrt[2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}","\frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}",1,"(Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[2 - 3*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d","A",2,2,33,0.06061,1,"{2995, 2994}"
444,1,95,0,0.1909731,"\int \frac{1+\cos (c+d x)}{\sqrt{-2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\frac{\sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{d}","\frac{\sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{d}",1,"(Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/d","A",2,2,33,0.06061,1,"{2995, 2994}"
445,1,72,0,0.0825297,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx","Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]),x]","\frac{2 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{3 d}","\frac{2 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{3 d}",1,"(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)","A",1,1,33,0.03030,1,"{2994}"
446,1,74,0,0.0993586,"\int \frac{1+\cos (c+d x)}{\sqrt{3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\frac{2 \sqrt{5} \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}","\frac{2 \sqrt{5} \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}",1,"(2*Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)","A",1,1,33,0.03030,1,"{2994}"
447,1,98,0,0.2015628,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx","Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]),x]","-\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}","-\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}",1,"(-2*Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)","A",2,2,33,0.06061,1,"{2995, 2994}"
448,1,96,0,0.1889412,"\int \frac{1+\cos (c+d x)}{\sqrt{-3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(1 + Cos[c + d*x])/(Sqrt[-3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{3 d}","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{3 d}",1,"(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)","A",2,2,33,0.06061,1,"{2995, 2994}"
449,0,0,0,0.0822421,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx","\text{Int}\left((A+B \cos (e+f x)) (c \cos (e+f x))^m (a+b \cos (e+f x))^n,x\right)",0,"Defer[Int][(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]), x]","A",0,0,0,0,-1,"{}"
450,1,595,0,1.9835676,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x]),x]","-\frac{\sin (e+f x) \left(4 a^3 A b \left(m^2+8 m+15\right)+6 a^2 b^2 B \left(m^2+7 m+10\right)+a^4 B \left(m^2+8 m+15\right)+4 a A b^3 \left(m^2+7 m+10\right)+b^4 B \left(m^2+6 m+8\right)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) (m+5) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(6 a^2 A b^2 \left(m^2+5 m+4\right)+a^4 A \left(m^2+6 m+8\right)+4 a^3 b B \left(m^2+5 m+4\right)+4 a b^3 B \left(m^2+4 m+3\right)+A b^4 \left(m^2+4 m+3\right)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) \left(a^2 A b \left(5 m^2+47 m+110\right)+2 a^3 B \left(m^2+10 m+28\right)+4 a b^2 B \left(m^2+8 m+15\right)+A b^3 \left(m^2+8 m+15\right)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4) (m+5)}+\frac{b^2 \sin (e+f x) \cos (e+f x) \left(a^2 B \left(m^2+11 m+36\right)+2 a A b (m+5)^2+b^2 B (m+4)^2\right) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4) (m+5)}+\frac{b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4) (m+5)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}","-\frac{\sin (e+f x) \left(4 a^3 A b \left(m^2+8 m+15\right)+6 a^2 b^2 B \left(m^2+7 m+10\right)+a^4 B \left(m^2+8 m+15\right)+4 a A b^3 \left(m^2+7 m+10\right)+b^4 B \left(m^2+6 m+8\right)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) (m+5) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(6 a^2 A b^2 \left(m^2+5 m+4\right)+a^4 A \left(m^2+6 m+8\right)+4 a^3 b B \left(m^2+5 m+4\right)+4 a b^3 B \left(m^2+4 m+3\right)+A b^4 \left(m^2+4 m+3\right)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) \left(a^2 A b \left(5 m^2+47 m+110\right)+2 a^3 B \left(m^2+10 m+28\right)+4 a b^2 B \left(m^2+8 m+15\right)+A b^3 \left(m^2+8 m+15\right)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4) (m+5)}+\frac{b^2 \sin (e+f x) \cos (e+f x) \left(a^2 B \left(m^2+11 m+36\right)+2 a A b (m+5)^2+b^2 B (m+4)^2\right) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4) (m+5)}+\frac{b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4) (m+5)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}",1,"(b*(A*b^3*(15 + 8*m + m^2) + 4*a*b^2*B*(15 + 8*m + m^2) + 2*a^3*B*(28 + 10*m + m^2) + a^2*A*b*(110 + 47*m + 5*m^2))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(4 + m)*(5 + m)) + (b^2*(b^2*B*(4 + m)^2 + 2*a*A*b*(5 + m)^2 + a^2*B*(36 + 11*m + m^2))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(3 + m)*(4 + m)*(5 + m)) + (b*(A*b*(5 + m) + a*B*(8 + m))*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^2*Sin[e + f*x])/(c*f*(4 + m)*(5 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^3*Sin[e + f*x])/(c*f*(5 + m)) - ((A*b^4*(3 + 4*m + m^2) + 4*a*b^3*B*(3 + 4*m + m^2) + 6*a^2*A*b^2*(4 + 5*m + m^2) + 4*a^3*b*B*(4 + 5*m + m^2) + a^4*A*(8 + 6*m + m^2))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^4*B*(8 + 6*m + m^2) + 4*a*A*b^3*(10 + 7*m + m^2) + 6*a^2*b^2*B*(10 + 7*m + m^2) + 4*a^3*A*b*(15 + 8*m + m^2) + a^4*B*(15 + 8*m + m^2))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*(5 + m)*Sqrt[Sin[e + f*x]^2])","A",7,6,33,0.1818,1,"{2990, 3049, 3033, 3023, 2748, 2643}"
451,1,406,0,1.0533968,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x]),x]","-\frac{\sin (e+f x) \left(3 a^2 A b (m+3)+a^3 B (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(b (m+1) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right)+a^2 (m+2) (a A (m+4)+b B (m+1))\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4)}+\frac{b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}","-\frac{\sin (e+f x) \left(3 a^2 A b (m+3)+a^3 B (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(b (m+1) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right)+a^2 (m+2) (a A (m+4)+b B (m+1))\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4)}+\frac{b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}",1,"(b*(b^2*B*(3 + m) + 3*a*A*b*(4 + m) + 2*a^2*B*(5 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(4 + m)) + (b^2*(A*b*(4 + m) + a*B*(6 + m))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(3 + m)*(4 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^2*Sin[e + f*x])/(c*f*(4 + m)) - ((a^2*(2 + m)*(b*B*(1 + m) + a*A*(4 + m)) + b*(1 + m)*(b^2*B*(3 + m) + 3*a*A*b*(4 + m) + 2*a^2*B*(5 + m)))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[e + f*x]^2]) - ((A*b^3*(2 + m) + 3*a*b^2*B*(2 + m) + 3*a^2*A*b*(3 + m) + a^3*B*(3 + m))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e + f*x]^2])","A",6,5,33,0.1515,1,"{2990, 3033, 3023, 2748, 2643}"
452,1,287,0,0.5411137,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x]),x]","-\frac{\sin (e+f x) \left(a^2 A (m+2)+2 a b B (m+1)+A b^2 (m+1)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(a (m+3) (a B+2 A b)+b^2 B (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+4)+A b (m+3)) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+3)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x)) (c \cos (e+f x))^{m+1}}{c f (m+3)}","-\frac{\sin (e+f x) \left(a^2 A (m+2)+2 a b B (m+1)+A b^2 (m+1)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(a (m+3) (a B+2 A b)+b^2 B (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+4)+A b (m+3)) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+3)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x)) (c \cos (e+f x))^{m+1}}{c f (m+3)}",1,"(b*(A*b*(3 + m) + a*B*(4 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(3 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])*Sin[e + f*x])/(c*f*(3 + m)) - ((A*b^2*(1 + m) + 2*a*b*B*(1 + m) + a^2*A*(2 + m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^2*B*(2 + m) + a*(2*A*b + a*B)*(3 + m))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e + f*x]^2])","A",5,4,33,0.1212,1,"{2990, 3023, 2748, 2643}"
453,1,196,0,0.2463983,"\int (c \cos (e+f x))^m (a+b \cos (e+f x)) (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])*(A + B*Cos[e + f*x]),x]","-\frac{(a B+A b) \sin (e+f x) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) (a A (m+2)+b B (m+1)) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}+\frac{b B \sin (e+f x) (c \cos (e+f x))^{m+1}}{c f (m+2)}","-\frac{(a B+A b) \sin (e+f x) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) (a A (m+2)+b B (m+1)) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}+\frac{b B \sin (e+f x) (c \cos (e+f x))^{m+1}}{c f (m+2)}",1,"(b*B*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)) - ((b*B*(1 + m) + a*A*(2 + m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((A*b + a*B)*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*Sqrt[Sin[e + f*x]^2])","A",5,4,31,0.1290,1,"{2968, 3023, 2748, 2643}"
454,1,286,0,0.4123377,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{a+b \cos (e+f x)} \, dx","Int[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/(a + b*Cos[e + f*x]),x]","\frac{a c (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b f \left(a^2-b^2\right)}-\frac{(A b-a B) \sin (e+f x) \cos ^2(e+f x)^{-m/2} (c \cos (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{B \sin (e+f x) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{b c f (m+1) \sqrt{\sin ^2(e+f x)}}","\frac{a c (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b f \left(a^2-b^2\right)}-\frac{(A b-a B) \sin (e+f x) \cos ^2(e+f x)^{-m/2} (c \cos (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{B \sin (e+f x) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{b c f (m+1) \sqrt{\sin ^2(e+f x)}}",1,"(a*(A*b - a*B)*c*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(c*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Sin[e + f*x])/(b*(a^2 - b^2)*f) - ((A*b - a*B)*AppellF1[1/2, -m/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(c*Cos[e + f*x])^m*Sin[e + f*x])/((a^2 - b^2)*f*(Cos[e + f*x]^2)^(m/2)) - (B*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(b*c*f*(1 + m)*Sqrt[Sin[e + f*x]^2])","A",7,5,33,0.1515,1,"{3002, 2643, 2823, 3189, 429}"
455,0,0,0,0.5265524,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx","\frac{2 \text{Int}\left(\frac{(c \cos (e+f x))^m \left(\frac{1}{2} c \cos (e+f x) \left(a (2 m+5) (a B+2 A b)+b^2 B (2 m+3)\right)+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (m+3)+A b (2 m+5))+\frac{1}{2} a c \left(2 a A \left(m+\frac{5}{2}\right)+2 b B (m+1)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{c (2 m+5)}+\frac{2 b B \sin (e+f x) \sqrt{a+b \cos (e+f x)} (c \cos (e+f x))^{m+1}}{c f (2 m+5)}",0,"(2*b*B*(c*Cos[e + f*x])^(1 + m)*Sqrt[a + b*Cos[e + f*x]]*Sin[e + f*x])/(c*f*(5 + 2*m)) + (2*Defer[Int][((c*Cos[e + f*x])^m*((a*c*(2*b*B*(1 + m) + 2*a*A*(5/2 + m)))/2 + (c*(b^2*B*(3 + 2*m) + a*(2*A*b + a*B)*(5 + 2*m))*Cos[e + f*x])/2 + (b*c*(2*a*B*(3 + m) + A*b*(5 + 2*m))*Cos[e + f*x]^2)/2))/Sqrt[a + b*Cos[e + f*x]], x])/(c*(5 + 2*m))","A",0,0,0,0,-1,"{}"
456,0,0,0,0.1162547,"\int (c \cos (e+f x))^m \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx","Int[(c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx","\text{Int}\left(\sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \cos (e+f x))^m,x\right)",0,"Defer[Int][(c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]), x]","A",0,0,0,0,-1,"{}"
457,0,0,0,0.1230473,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx","Int[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]],x]","\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx","\text{Int}\left(\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^m}{\sqrt{a+b \cos (e+f x)}},x\right)",0,"Defer[Int][((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]","A",0,0,0,0,-1,"{}"
458,0,0,0,0.4997438,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx","Int[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/(a + b*Cos[e + f*x])^(3/2),x]","\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx","\frac{2 \text{Int}\left(\frac{(c \cos (e+f x))^m \left(-\frac{1}{2} b c (2 m+3) (A b-a B) \cos ^2(e+f x)-\frac{1}{2} a c (A b-a B) \cos (e+f x)+\frac{1}{2} c \left(2 b \left(m+\frac{1}{2}\right) (A b-a B)+a (a A-b B)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{a c \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (e+f x) (c \cos (e+f x))^{m+1}}{a c f \left(a^2-b^2\right) \sqrt{a+b \cos (e+f x)}}",0,"(2*b*(A*b - a*B)*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*(a^2 - b^2)*c*f*Sqrt[a + b*Cos[e + f*x]]) + (2*Defer[Int][((c*Cos[e + f*x])^m*((c*(a*(a*A - b*B) + 2*b*(A*b - a*B)*(1/2 + m)))/2 - (a*(A*b - a*B)*c*Cos[e + f*x])/2 - (b*(A*b - a*B)*c*(3 + 2*m)*Cos[e + f*x]^2)/2))/Sqrt[a + b*Cos[e + f*x]], x])/(a*(a^2 - b^2)*c)","A",0,0,0,0,-1,"{}"
459,1,172,0,0.2220717,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,7,31,0.2258,1,"{2960, 3997, 3787, 3768, 3771, 2639, 2641}"
460,1,135,0,0.1921171,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,31,0.2258,1,"{2960, 3997, 3787, 3771, 2641, 3768, 2639}"
461,1,106,0,0.1788317,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}","\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(-2*a*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,31,0.1935,1,"{2960, 3997, 3787, 3771, 2639, 2641}"
462,1,110,0,0.1839175,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 a (3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,31,0.1935,1,"{2960, 3996, 3787, 3771, 2639, 2641}"
463,1,141,0,0.2026546,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 a (A+B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a (A+B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*a*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,31,0.2258,1,"{2960, 3996, 3787, 3769, 3771, 2641, 2639}"
464,1,172,0,0.2241613,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 a (A+B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a (A+B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,31,0.2258,1,"{2960, 3996, 3787, 3769, 3771, 2639, 2641}"
465,1,199,0,0.3452334,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}","\frac{2 a^2 (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{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)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(7*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",9,8,33,0.2424,1,"{2960, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
466,1,160,0,0.3181985,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a^2 (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (2 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{3 d}-\frac{4 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 a^2 (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (2 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{3 d}-\frac{4 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-4*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{2960, 4018, 3997, 3787, 3771, 2639, 2641}"
467,1,160,0,0.322774,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (3 A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (3 A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(4*a^2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(3*A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(3*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{2960, 4017, 3997, 3787, 3771, 2639, 2641}"
468,1,166,0,0.3402045,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (5 A+7 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (2 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (5 A+4 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 (5 A+7 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (2 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (5 A+4 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(4*a^2*(5*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",8,7,33,0.2121,1,"{2960, 4017, 3996, 3787, 3771, 2639, 2641}"
469,1,201,0,0.3691145,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 (7 A+9 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (7 A+9 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (4 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(7*A + 9*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",9,8,33,0.2424,1,"{2960, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
470,1,244,0,0.5051576,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (11 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d}+\frac{4 a^3 (7 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (13 A+21 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}","\frac{4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (11 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d}+\frac{4 a^3 (7 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (13 A+21 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}",1,"(-4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(7*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(41*A + 42*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*A*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(11*A + 7*B)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d)","A",10,8,33,0.2424,1,"{2960, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
471,1,211,0,0.4945072,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{4 a^3 (21 A+20 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (9 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}","\frac{4 a^3 (21 A+20 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (9 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}",1,"(-4*a^3*(9*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(21*A + 20*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(9*A + 5*B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,7,33,0.2121,1,"{2960, 4018, 3997, 3787, 3771, 2639, 2641}"
472,1,199,0,0.4910885,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{3 d}+\frac{20 a^3 (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}","\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{3 d}+\frac{20 a^3 (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(-4*a^3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(4*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(A - B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",9,8,33,0.2424,1,"{2960, 4017, 4018, 3997, 3787, 3771, 2639, 2641}"
473,1,211,0,0.4949713,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (5 A+9 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (5 A+9 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(4*a^3*(5*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A - 6*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*A + 9*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",9,7,33,0.2121,1,"{2960, 4017, 3997, 3787, 3771, 2639, 2641}"
474,1,211,0,0.5075215,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 (7 A+11 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (9 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 (7 A+11 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (9 A+7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*a^3*(9*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",9,7,33,0.2121,1,"{2960, 4017, 3996, 3787, 3771, 2639, 2641}"
475,1,244,0,0.5353393,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 A+13 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 A+13 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(24*A + 23*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^3*(13*A + 11*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*A + 13*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",10,8,33,0.2424,1,"{2960, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
476,1,193,0,0.3041573,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]),x]","-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(5 A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(5 A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*(A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,7,33,0.2121,1,"{2960, 4019, 3787, 3768, 3771, 2639, 2641}"
477,1,159,0,0.2758706,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]),x]","-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((3*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,7,33,0.2121,1,"{2960, 4019, 3787, 3771, 2641, 3768, 2639}"
478,1,123,0,0.2446078,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{2960, 4019, 3787, 3771, 2639, 2641}"
479,1,125,0,0.2529129,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-(((A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{2960, 4020, 3787, 3771, 2639, 2641}"
480,1,163,0,0.2807199,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}-\frac{(3 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}-\frac{(3 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A - 5*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","A",8,7,33,0.2121,1,"{2960, 4020, 3787, 3769, 3771, 2641, 2639}"
481,1,196,0,0.300342,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{5 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}","\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{5 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-3*(5*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((5*A - 7*B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",9,7,33,0.2121,1,"{2960, 4020, 3787, 3769, 3771, 2639, 2641}"
482,1,208,0,0.4307575,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,x]","-\frac{(5 A-2 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(5 A-2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{(5 A-2 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(5 A-2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - 2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,7,33,0.2121,1,"{2960, 4019, 3787, 3771, 2641, 3768, 2639}"
483,1,161,0,0.3880558,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,x]","\frac{(2 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{(2 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,6,33,0.1818,1,"{2960, 4019, 3787, 3771, 2639, 2641}"
484,1,168,0,0.3910554,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{(A+2 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}","\frac{(A+2 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(A+2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"-((B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A + 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,33,0.2121,1,"{2960, 4019, 4020, 3787, 3771, 2639, 2641}"
485,1,176,0,0.4062329,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{(2 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}","\frac{(2 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"-(((A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((2*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,6,33,0.1818,1,"{2960, 4020, 3787, 3771, 2639, 2641}"
486,1,206,0,0.4339143,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","-\frac{5 (A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{(4 A-7 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}-\frac{5 (A-2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}","-\frac{5 (A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{(4 A-7 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}-\frac{5 (A-2 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-7 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"((4*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (5*(A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + ((4*A - 7*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",9,7,33,0.2121,1,"{2960, 4020, 3787, 3769, 3771, 2641, 2639}"
487,1,261,0,0.6121149,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3,x]","-\frac{(13 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(8 A-3 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}","-\frac{(13 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(8 A-3 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-((49*A - 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",10,7,33,0.2121,1,"{2960, 4019, 3787, 3771, 2641, 3768, 2639}"
488,1,222,0,0.581253,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3,x]","-\frac{(9 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(6 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}","-\frac{(9 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(6 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"((9*A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((6*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((9*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{2960, 4019, 3787, 3771, 2639, 2641}"
489,1,216,0,0.566247,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{(A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}","\frac{(A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((4*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,33,0.2121,1,"{2960, 4019, 4020, 3787, 3771, 2639, 2641}"
490,1,222,0,0.5799671,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{(A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(2 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}","\frac{(A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(2 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"-((A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,33,0.2121,1,"{2960, 4019, 4020, 3787, 3771, 2639, 2641}"
491,1,228,0,0.5766932,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{(3 A-13 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A-13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(3 A-8 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}","\frac{(3 A-13 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A-13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(3 A-8 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"-((9*A - 49*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A - 8*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A - 13*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,6,33,0.1818,1,"{2960, 4020, 3787, 3771, 2639, 2641}"
492,1,259,0,0.6235467,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","-\frac{(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}+\frac{7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-2 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}","-\frac{(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}+\frac{7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-2 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((13*A - 33*B)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",10,7,33,0.2121,1,"{2960, 4020, 3787, 3769, 3771, 2641, 2639}"
493,1,220,0,0.486495,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(32*a*(8*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(8*A + 9*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(8*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])","A",6,4,35,0.1143,1,"{2961, 2980, 2772, 2771}"
494,1,175,0,0.4055952,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(16*a*(6*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])","A",5,4,35,0.1143,1,"{2961, 2980, 2772, 2771}"
495,1,130,0,0.3339995,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{2961, 2980, 2772, 2771}"
496,1,85,0,0.2661357,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a (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 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a (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 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*(2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,35,0.08571,1,"{2961, 2980, 2771}"
497,1,96,0,0.2727468,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} 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)}{d}","\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} 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)}{d}",1,"(2*Sqrt[a]*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]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,35,0.1143,1,"{2961, 2980, 2774, 216}"
498,1,98,0,0.271111,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (2 A+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)}{d}+\frac{a B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (2 A+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)}{d}+\frac{a B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(2*A + 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]])/d + (a*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",4,4,35,0.1143,1,"{2961, 2981, 2774, 216}"
499,1,151,0,0.3375944,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} (4 A+3 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)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} (4 A+3 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)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(4*A + 3*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]])/(4*d) + (a*B*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(4*A + 3*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 2981, 2770, 2774, 216}"
500,1,196,0,0.413737,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a (6 A+5 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 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)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","\frac{a (6 A+5 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 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)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*(6*A + 5*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]])/(8*d) + (a*B*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2961, 2981, 2770, 2774, 216}"
501,1,275,0,0.723155,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (12 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^2 (168 A+187 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}","\frac{2 a^2 (12 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^2 (168 A+187 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(32*a^2*(168*A + 187*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(168*A + 187*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(168*A + 187*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(168*A + 187*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(12*A + 11*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
502,1,228,0,0.6499731,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (10 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^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}","\frac{2 a^2 (10 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^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(16*a^2*(34*A + 39*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*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])/(9*d)","A",6,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
503,1,181,0,0.5608344,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (8 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^2 (52 A+63 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}","\frac{2 a^2 (8 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^2 (52 A+63 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(4*a^2*(52*A + 63*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*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])/(7*d)","A",5,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
504,1,134,0,0.4689834,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(2*a^2*(18*A + 25*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(6*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",4,4,35,0.1143,1,"{2961, 2975, 2980, 2771}"
505,1,145,0,0.4507016,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} 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)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} 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)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(2*a^(3/2)*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]])/d + (2*a^2*(4*A + 3*B)*Sqrt[Sec[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]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{2961, 2975, 2980, 2774, 216}"
506,1,146,0,0.4658382,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{a^{3/2} (2 A+3 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)}{d}-\frac{a^2 (2 A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}","\frac{a^{3/2} (2 A+3 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)}{d}-\frac{a^2 (2 A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}",1,"(a^(3/2)*(2*A + 3*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]])/d - (a^2*(2*A - B)*Sin[c + d*x])/(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","A",5,5,35,0.1429,1,"{2961, 2975, 2981, 2774, 216}"
507,1,153,0,0.4554536,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{a^{3/2} (12 A+7 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)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (12 A+7 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)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}",1,"(a^(3/2)*(12*A + 7*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]])/(4*d) + (a^2*(4*A + 5*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 2976, 2981, 2774, 216}"
508,1,200,0,0.5421702,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 (6 A+7 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (14 A+11 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)}{8 d}+\frac{a^2 (14 A+11 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^2 (6 A+7 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (14 A+11 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)}{8 d}+\frac{a^2 (14 A+11 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(3/2)*(14*A + 11*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]])/(8*d) + (a^2*(6*A + 7*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a^2*(14*A + 11*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 2976, 2981, 2770, 2774, 216}"
509,1,247,0,0.6436384,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a^2 (88 A+75 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+9 B) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (88 A+75 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)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^2 (88 A+75 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+9 B) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (88 A+75 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)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(3/2)*(88*A + 75*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]])/(64*d) + (a^2*(8*A + 9*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2976, 2981, 2770, 2774, 216}"
510,1,322,0,0.9388887,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{15}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(15/2),x]","\frac{2 a^2 (16 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{143 d}+\frac{2 a^3 (280 A+299 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{1287 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^3 (4184 A+4615 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{13 d}","\frac{2 a^2 (16 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{143 d}+\frac{2 a^3 (280 A+299 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{1287 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^3 (4184 A+4615 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{13 d}",1,"(32*a^3*(4184*A + 4615*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(280*A + 299*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(16*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)","A",8,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
511,1,275,0,0.8482687,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2),x]","\frac{2 a^2 (14 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^3 (194 A+209 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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 (14 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^3 (194 A+209 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3*(710*A + 803*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(194*A + 209*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*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 + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",7,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
512,1,228,0,0.7676704,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 (4 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^3 (124 A+135 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (292 A+345 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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 (4 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^3 (124 A+135 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (292 A+345 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3*(292*A + 345*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(124*A + 135*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*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 + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,35,0.1429,1,"{2961, 2975, 2980, 2772, 2771}"
513,1,181,0,0.6761078,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 (10 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^3 (10 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (230 A+301 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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 (10 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^3 (10 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (230 A+301 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a 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^3*(230*A + 301*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(10*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*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 + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,4,35,0.1143,1,"{2961, 2975, 2980, 2771}"
514,1,192,0,0.6229236,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 (8 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^3 (32 A+35 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} 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)}{d}+\frac{2 a 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 (8 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^3 (32 A+35 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{5/2} 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)}{d}+\frac{2 a 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^(5/2)*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]])/d + (2*a^3*(32*A + 35*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*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 + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,5,35,0.1429,1,"{2961, 2975, 2980, 2774, 216}"
515,1,193,0,0.6541432,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{a^{5/2} (2 A+5 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)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{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{a^{5/2} (2 A+5 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)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{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}",1,"(a^(5/2)*(2*A + 5*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]])/d - (a^3*(14*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",6,5,35,0.1429,1,"{2961, 2975, 2981, 2774, 216}"
516,1,198,0,0.6644387,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{a^{5/2} (20 A+19 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)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}","\frac{a^{5/2} (20 A+19 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)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^(5/2)*(20*A + 19*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]])/(4*d) - (a^3*(4*A - 9*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(4*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,35,0.1714,1,"{2961, 2975, 2976, 2981, 2774, 216}"
517,1,200,0,0.6503323,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{a^{5/2} (38 A+25 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)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (38 A+25 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)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^(5/2)*(38*A + 25*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]])/(8*d) + (a^3*(54*A + 49*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2961, 2976, 2981, 2774, 216}"
518,1,247,0,0.7516285,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 (104 A+95 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (200 A+163 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)}{64 d}+\frac{a^3 (200 A+163 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{a^3 (104 A+95 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^{5/2} (200 A+163 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)}{64 d}+\frac{a^3 (200 A+163 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^(5/2)*(200*A + 163*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]])/(64*d) + (a^3*(104*A + 95*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(8*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^3*(200*A + 163*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2976, 2981, 2770, 2774, 216}"
519,1,294,0,0.8655671,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a^3 (326 A+283 B) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (170 A+157 B) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (326 A+283 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)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{a^3 (326 A+283 B) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (170 A+157 B) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (326 A+283 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)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^(5/2)*(326*A + 283*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]])/(128*d) + (a^3*(170*A + 157*B)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(10*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,6,35,0.1714,1,"{2961, 2976, 2981, 2770, 2774, 216}"
520,1,295,0,1.0565528,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\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 (19 A-3 B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (19 A-3 B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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)*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)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B)*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",9,5,35,0.1429,1,"{2961, 2984, 12, 2782, 205}"
521,1,250,0,0.8379509,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\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 (31 A-7 B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (31 A-7 B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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)*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)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B)*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",8,5,35,0.1429,1,"{2961, 2984, 12, 2782, 205}"
522,1,207,0,0.6468251,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\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 (13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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)*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)*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",7,5,35,0.1429,1,"{2961, 2984, 12, 2782, 205}"
523,1,162,0,0.4529879,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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)*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",6,5,35,0.1429,1,"{2961, 2984, 12, 2782, 205}"
524,1,119,0,0.3061329,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"-((Sqrt[2]*(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)) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 2984, 12, 2782, 205}"
525,1,140,0,0.3496686,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} (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{2 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) \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 \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*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)*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)","A",6,6,35,0.1714,1,"{2961, 2982, 2782, 205, 2774, 216}"
526,1,181,0,0.5125428,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{(2 A-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) \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(2 A-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) \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((2*A - 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)*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*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{2961, 2983, 2982, 2782, 205, 2774, 216}"
527,1,230,0,0.6991749,"\int \frac{A+B \cos (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])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","-\frac{(4 A-7 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)}{4 \sqrt{a} d}+\frac{(4 A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}","-\frac{(4 A-7 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)}{4 \sqrt{a} d}+\frac{(4 A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((4*A - 7*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]])/(4*Sqrt[a]*d) + (Sqrt[2]*(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*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A - B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,7,35,0.2000,1,"{2961, 2983, 2982, 2782, 205, 2774, 216}"
528,1,192,0,0.6652477,"\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}"
529,1,317,0,1.1060899,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(19 A-15 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A-7 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}","\frac{(19 A-15 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A-7 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}",1,"((19*A - 15*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]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
530,1,270,0,0.8870789,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(15 A-11 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}","-\frac{(15 A-11 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}",1,"-((15*A - 11*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]])/(2*Sqrt[2]*a^(3/2)*d) + ((147*A - 95*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
531,1,223,0,0.7018275,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"((11*A - 7*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]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
532,1,176,0,0.5192388,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-((7*A - 3*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]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
533,1,127,0,0.3372858,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"((3*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]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 2978, 12, 2782, 205}"
534,1,185,0,0.5438159,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 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)}{a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 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)}{a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*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]])/(a^(3/2)*d) + ((A - 5*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]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,7,35,0.2000,1,"{2961, 2977, 2982, 2782, 205, 2774, 216}"
535,1,237,0,0.7364215,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{(2 A-3 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)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{(2 A-3 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)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((2*A - 3*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]])/(a^(3/2)*d) - ((5*A - 9*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]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((A - 3*B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,35,0.2286,1,"{2961, 2977, 2983, 2982, 2782, 205, 2774, 216}"
536,1,317,0,1.1247804,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(157 A-85 B) \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) \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) \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) \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) \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) \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) \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) \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) \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) \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) \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) \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)*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)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B)*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)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
537,1,270,0,0.9287024,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(95 A-39 B) \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) \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) \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) \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) \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) \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) \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) \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) \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) \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)*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)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*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)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
538,1,223,0,0.7291199,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-((75*A - 19*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]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
539,1,176,0,0.526173,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((19*A + 5*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]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,5,35,0.1429,1,"{2961, 2978, 12, 2782, 205}"
540,1,174,0,0.5109079,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((5*A + 3*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]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + ((A + 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 2977, 2978, 12, 2782, 205}"
541,1,234,0,0.7381464,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 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)}{a^{5/2} d}+\frac{(A-B) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 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)}{a^{5/2} d}+\frac{(A-B) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*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]])/(a^(5/2)*d) + ((3*A - 43*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]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,7,35,0.2000,1,"{2961, 2977, 2982, 2782, 205, 2774, 216}"
542,1,286,0,0.9814936,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{(2 A-5 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)}{a^{5/2} d}-\frac{(11 A-35 B) \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) \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{(7 A-15 B) \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) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{(2 A-5 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)}{a^{5/2} d}-\frac{(11 A-35 B) \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) \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{(7 A-15 B) \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) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((2*A - 5*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]])/(a^(5/2)*d) - ((43*A - 115*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]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((11*A - 35*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,8,35,0.2286,1,"{2961, 2977, 2983, 2982, 2782, 205, 2774, 216}"
543,1,317,0,1.1466523,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{(579 A-199 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}","\frac{(579 A-199 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"((1015*A - 363*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]])/(64*Sqrt[2]*a^(7/2)*d) - ((1887*A - 691*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",9,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
544,1,270,0,0.9499555,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{(691 A-103 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","\frac{(691 A-103 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(-3*(121*A - 21*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]])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,35,0.1714,1,"{2961, 2978, 2984, 12, 2782, 205}"
545,1,223,0,0.727955,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(7/2),x]","-\frac{(103 A+5 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","-\frac{(103 A+5 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"((63*A + 13*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]])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - ((5*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((103*A + 5*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,5,35,0.1429,1,"{2961, 2978, 12, 2782, 205}"
546,1,221,0,0.7250665,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]),x]","-\frac{(5 A-17 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A+3 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","-\frac{(5 A-17 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A+3 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"((13*A + 7*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]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + ((A + 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((5*A - 17*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2977, 2978, 12, 2782, 205}"
547,1,221,0,0.7231352,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)),x]","\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((7*A + 5*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]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) + ((A - 13*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + ((17*A + 67*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,6,35,0.1714,1,"{2961, 2977, 2978, 12, 2782, 205}"
548,1,281,0,0.9191238,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)),x]","\frac{(5 A-49 B) \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 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)}{a^{7/2} d}+\frac{(5 A-17 B) \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","\frac{(5 A-49 B) \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 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)}{a^{7/2} d}+\frac{(5 A-17 B) \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(2*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]])/(a^(7/2)*d) + ((5*A - 177*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]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + ((5*A - 17*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 49*B)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",9,7,35,0.2000,1,"{2961, 2977, 2982, 2782, 205, 2774, 216}"
549,1,333,0,1.2049752,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]","\frac{(79 A-259 B) \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(2 A-7 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)}{a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(3 A-7 B) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","\frac{(79 A-259 B) \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(2 A-7 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)}{a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(3 A-7 B) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"((2*A - 7*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]])/(a^(7/2)*d) - ((177*A - 637*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]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) + ((3*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((79*A - 259*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - (7*(7*A - 27*B)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",10,8,35,0.2286,1,"{2961, 2977, 2983, 2982, 2782, 205, 2774, 216}"
550,1,180,0,0.2233256,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (3 a A+5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (3 a A+5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(-2*(3*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a*A + 5*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(A*b + a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,7,31,0.2258,1,"{2960, 3997, 3787, 3768, 3771, 2639, 2641}"
551,1,143,0,0.2010434,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 (a A+3 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 (a A+3 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*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",8,7,31,0.2258,1,"{2960, 3997, 3787, 3771, 2641, 3768, 2639}"
552,1,111,0,0.1804136,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a A-b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}","\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a A-b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(-2*(a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,31,0.1935,1,"{2960, 3997, 3787, 3771, 2639, 2641}"
553,1,115,0,0.1877896,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 (3 a A+b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 (3 a A+b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,31,0.1935,1,"{2960, 3996, 3787, 3771, 2639, 2641}"
554,1,148,0,0.2082883,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 (a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a A+3 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 (a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a A+3 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,31,0.2258,1,"{2960, 3996, 3787, 3769, 3771, 2641, 2639}"
555,1,180,0,0.2338846,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 (a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 a A+5 b B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 (a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 a A+5 b B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a A+5 b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*a*A + 5*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,31,0.2258,1,"{2960, 3996, 3787, 3769, 3771, 2639, 2641}"
556,1,221,0,0.3798521,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 a B+7 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 a B+7 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}",1,"(-2*(3*a^2*A + 5*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a^2*A + 5*b*(A*b + 2*a*B))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",9,8,33,0.2424,1,"{2960, 4026, 4047, 3768, 3771, 2639, 4046, 2641}"
557,1,177,0,0.3516158,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}",1,"(-2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A*b + 3*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,33,0.2121,1,"{2960, 4026, 4047, 3771, 2641, 4046, 2639}"
558,1,161,0,0.3201265,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 A-b (2 a B+A b)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 A-b (2 a B+A b)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b^2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*(a^2*A - b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,7,33,0.2121,1,"{2960, 4024, 4047, 3771, 2641, 4046, 2639}"
559,1,171,0,0.3364871,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a^2 B+10 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a^2 B+10 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{2960, 4024, 4047, 3771, 2639, 4045, 2641}"
560,1,213,0,0.372137,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,8,33,0.2424,1,"{2960, 4024, 4047, 3769, 3771, 2641, 4045, 2639}"
561,1,295,0,0.5972258,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 a \left(5 a^2 A+21 a b B+18 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}","\frac{2 a \left(5 a^2 A+21 a b B+18 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(9 a^2 A b+3 a^3 B+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}",1,"(-2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*a^2*A + 18*A*b^2 + 21*a*b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(11*A*b + 7*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)","A",10,9,33,0.2727,1,"{2960, 4026, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
562,1,244,0,0.5770689,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a \left(3 a^2 A+15 a b B+14 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}{5 d}","\frac{2 a \left(3 a^2 A+15 a b B+14 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}{5 d}",1,"(-2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*a^2*A + 14*A*b^2 + 15*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(9*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)","A",9,8,33,0.2424,1,"{2960, 4026, 4076, 4047, 3771, 2641, 4046, 2639}"
563,1,239,0,0.565625,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a \left(3 a^2 B+9 a A b-2 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (a A-b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a \left(3 a^2 B+9 a A b-2 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (a A-b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(9*a*A*b + 3*a^2*B - 2*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a*A - b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",9,8,33,0.2424,1,"{2960, 4025, 4076, 4047, 3771, 2641, 4046, 2639}"
564,1,237,0,0.5329473,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \left(9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*(5*A*b + 9*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(5*a*A - b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",9,8,33,0.2424,1,"{2960, 4025, 4074, 4047, 3771, 2641, 4046, 2639}"
565,1,245,0,0.5438587,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",9,8,33,0.2424,1,"{2960, 4025, 4074, 4047, 3771, 2639, 4045, 2641}"
566,1,295,0,0.578549,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^2 A b+7 a^3 B+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(9*A*b + 13*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",10,9,33,0.2727,1,"{2960, 4025, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
567,1,210,0,0.8078387,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","-\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 b (A b-a B) \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 \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 (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 b (A b-a B) \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 \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*b*(A*b - a*B)*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",11,10,33,0.3030,1,"{2960, 4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
568,1,126,0,0.458501,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","-\frac{2 (A b-a B) \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 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 (A b-a B) \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 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}",1,"(-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",8,8,33,0.2424,1,"{2960, 4033, 4106, 3849, 2805, 12, 3771, 2639}"
569,1,101,0,0.2827835,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{2 (A b-a B) \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 d (a+b)}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (A b-a B) \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 d (a+b)}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d)","A",6,6,33,0.1818,1,"{2960, 4038, 3771, 2641, 3849, 2805}"
570,1,149,0,0.3462728,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \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 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) - (2*a*(A*b - a*B)*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",8,8,33,0.2424,1,"{2960, 4038, 3771, 2639, 3848, 2803, 2641, 2805}"
571,1,197,0,0.5597354,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","-\frac{2 \left(-3 a^2 B+3 a A b-b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}","-\frac{2 \left(-3 a^2 B+3 a A b-b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 a^2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"(2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) + (2*a^2*(A*b - a*B)*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*B*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])","A",10,9,33,0.2727,1,"{2960, 4034, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
572,1,405,0,1.2887194,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{\left(4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{\left(4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(4 a^2 A b-2 a^3 B+3 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 a^2 A b-5 a^3 B+3 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",12,10,33,0.3030,1,"{2960, 4029, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
573,1,316,0,0.9416325,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+a b B-3 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 A+a b B-3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 A b-3 a^3 B+a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+a b B-3 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 A+a b B-3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(5 a^2 A b-3 a^3 B+a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"-(((2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4029, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
574,1,260,0,0.6129913,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b+a^3 (-B)-a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b+a^3 (-B)-a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}",1,"-(((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{2960, 4029, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
575,1,258,0,0.6053121,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(a^2 B+a A b-2 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 A b+a^3 B-3 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(a^2 B+a A b-2 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(a^2 A b+a^3 B-3 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{2960, 4027, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
576,1,284,0,0.6584895,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(a^2 A b-3 a^3 B+4 a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 A b-3 a^3 B+5 a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(a^2 A b-3 a^3 B+4 a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(a^2 A b-3 a^3 B+5 a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"-(((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*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*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,33,0.2727,1,"{2960, 4030, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
577,1,363,0,0.9624765,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(9 a^3 A b+16 a^2 b^2 B-15 a^4 B-12 a A b^3+2 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b-5 a^3 B+4 a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 A b-5 a^3 B+7 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}","\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(9 a^3 A b+16 a^2 b^2 B-15 a^4 B-12 a A b^3+2 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2 A b-5 a^3 B+4 a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2 A b-5 a^3 B+7 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*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^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4030, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
578,1,480,0,1.4502892,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3,x]","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(11 a^2 A b-7 a^3 B+a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 A b^2+8 a^4 A+9 a^3 b B-3 a b^3 B+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-38 a^2 A b^3+35 a^4 A b+6 a^3 b^2 B-15 a^5 B-3 a b^4 B+15 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"-((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",12,11,33,0.3333,1,"{2960, 4029, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
579,1,405,0,1.0356626,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{\left(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+a b^4 B+3 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{\left(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(9 a^2 A b-5 a^3 B-a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(-6 a^2 A b^3+15 a^4 A b-10 a^3 b^2 B-3 a^5 B+a b^4 B+3 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d (a-b)^2 (a+b)^3}",1,"-((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4029, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
580,1,402,0,1.089677,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","-\frac{\left(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2 A b+a^3 (-B)-5 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-3 a b^4 B-A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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(7 a^2 A b-3 a^3 B-3 a b^2 B-A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2 A b+a^3 (-B)-5 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 A b^3+3 a^4 A b-10 a^3 b^2 B+a^5 B-3 a b^4 B-A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d (a-b)^2 (a+b)^3}",1,"((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4029, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
581,1,400,0,0.9550119,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\left(3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(a^3 A b-5 a^2 b^2 B+3 a^4 B-7 a A b^3+8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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(3 a^2 A b+a^3 B-7 a b^2 B+3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(a^3 A b-5 a^2 b^2 B+3 a^4 B-7 a A b^3+8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 A b+3 a^3 B-9 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 A b^3+a^4 A b-6 a^3 b^2 B+3 a^5 B+15 a b^4 B-3 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"-((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4027, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
582,1,427,0,1.0570069,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{a \left(a^2 A b-5 a^3 B+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(-5 a^2 A b^3+3 a^4 A b+33 a^3 b^2 B-15 a^5 B-24 a b^4 B+8 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 B-9 a A b^3-8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-35 a b^4 B+15 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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 \left(a^2 A b-5 a^3 B+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(-5 a^2 A b^3+3 a^4 A b+33 a^3 b^2 B-15 a^5 B-24 a b^4 B+8 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(3 a^3 A b+29 a^2 b^2 B-15 a^4 B-9 a A b^3-8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-6 a^2 A b^3+3 a^4 A b+38 a^3 b^2 B-15 a^5 B-35 a b^4 B+15 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*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) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,33,0.3030,1,"{2960, 4030, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
583,1,521,0,1.5287666,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{a \left(3 a^2 A b-7 a^3 B+13 a b^2 B-9 A b^3\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}+\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac{\left(15 a^3 A b+61 a^2 b^2 B-35 a^4 B-33 a A b^3-8 b^4 B\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{\left(-99 a^3 A b^3+45 a^5 A b+223 a^4 b^2 B-128 a^2 b^4 B-105 a^6 B+72 a A b^5-8 b^6 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-24 a b^4 B+8 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-63 a b^4 B+35 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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 \left(3 a^2 A b-7 a^3 B+13 a b^2 B-9 A b^3\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}+\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac{\left(15 a^3 A b+61 a^2 b^2 B-35 a^4 B-33 a A b^3-8 b^4 B\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{\left(-99 a^3 A b^3+45 a^5 A b+223 a^4 b^2 B-128 a^2 b^4 B-105 a^6 B+72 a A b^5-8 b^6 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 A b^3+15 a^4 A b+65 a^3 b^2 B-35 a^5 B-24 a b^4 B+8 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-38 a^2 A b^3+15 a^4 A b+86 a^3 b^2 B-35 a^5 B-63 a b^4 B+35 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 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*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*A*b - 99*a^3*A*b^3 + 72*a*A*b^5 - 105*a^6*B + 223*a^4*b^2*B - 128*a^2*b^4*B - 8*b^6*B)*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^2*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x]))","A",12,11,33,0.3333,1,"{2960, 4030, 4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
584,1,64,0,0.0371022,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),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 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*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",4,4,36,0.1111,1,"{21, 3768, 3771, 2641}"
585,1,60,0,0.0361397,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),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 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*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,36,0.1111,1,"{21, 3768, 3771, 2639}"
586,1,37,0,0.0234589,"\int \frac{(a B+b B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d","A",3,3,36,0.08333,1,"{21, 3771, 2641}"
587,1,37,0,0.0227405,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*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 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","A",3,3,36,0.08333,1,"{21, 3771, 2639}"
588,1,64,0,0.0390212,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),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 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}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",4,4,36,0.1111,1,"{21, 3769, 3771, 2641}"
589,1,64,0,0.036836,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),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 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}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",4,4,36,0.1111,1,"{21, 3769, 3771, 2639}"
590,1,473,0,1.437751,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+7 a b B-4 A b^2\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(a^2 (25 A-63 B)+2 a b (3 A-7 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 (a-b) \sqrt{a+b} \left(19 a^2 A b+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\cos (c+d x)} \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 (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 \left(25 a^2 A+7 a b B-4 A b^2\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(a^2 (25 A-63 B)+2 a b (3 A-7 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 (a-b) \sqrt{a+b} \left(19 a^2 A b+63 a^3 B-14 a b^2 B+8 A b^3\right) \sqrt{\cos (c+d x)} \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 (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]*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + a^2*(25*A - 63*B) + 2*a*b*(3*A - 7*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)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*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",7,6,35,0.1714,1,"{2961, 2999, 3055, 2998, 2816, 2994}"
591,1,390,0,1.0571062,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+5 a b 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}} E\left(\sin ^{-1}\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-5 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)}{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} \left(9 a^2 A+5 a b 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}} E\left(\sin ^{-1}\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-5 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)}{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]*(9*a^2*A - 2*A*b^2 + 5*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)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b - 5*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)])/(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",6,6,35,0.1714,1,"{2961, 2999, 3055, 2998, 2816, 2994}"
592,1,324,0,0.6751366,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 (a-b) \sqrt{a+b} (A-3 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 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 (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 (a-b) \sqrt{a+b} (A-3 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 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}",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*(a - b)*Sqrt[a + b]*(A - 3*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)])/(3*a*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",5,5,35,0.1429,1,"{2961, 2999, 2998, 2816, 2994}"
593,1,411,0,0.684636,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{a+b} (A b-a (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 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 d \sqrt{\sec (c+d x)}}-\frac{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}} \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} (A b-a (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 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 d \sqrt{\sec (c+d x)}}-\frac{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}} \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)*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*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(A*b - a*(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]*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)])/(d*Sqrt[Sec[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 2991, 2809, 2998, 2816, 2994}"
594,1,445,0,0.9026865,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sqrt[Sec[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",7,7,35,0.2000,1,"{2961, 3003, 3053, 2809, 2998, 2816, 2994}"
595,1,533,0,1.265078,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\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{(a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sqrt{a+b} (B (a+2 b)+4 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-b) \sqrt{a+b} (a B+4 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 a b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\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{(a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sqrt{a+b} (B (a+2 b)+4 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-b) \sqrt{a+b} (a B+4 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 a b d \sqrt{\sec (c+d x)}}+\frac{B \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*A*b + 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)])/(4*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(4*A*b + (a + 2*b)*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)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(4*a*A*b - a^2*B + 4*b^2*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)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)","A",8,8,35,0.2286,1,"{2961, 3003, 3061, 3053, 2809, 2998, 2816, 2994}"
596,1,620,0,1.740005,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\left(-3 a^2 B+6 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 B+6 a A b+16 b^2 B\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{\sqrt{a+b} \left(2 a^2 A b+a^3 (-B)-4 a b^2 B-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}} \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{\sqrt{a+b} (a+2 b) (-3 a B+6 A b+8 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}} F\left(\sin ^{-1}\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{(2 A b-a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}","\frac{\left(-3 a^2 B+6 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 B+6 a A b+16 b^2 B\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{\sqrt{a+b} \left(2 a^2 A b+a^3 (-B)-4 a b^2 B-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}} \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{\sqrt{a+b} (a+2 b) (-3 a B+6 A b+8 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}} F\left(\sin ^{-1}\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{(2 A b-a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{B \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]*(6*a*A*b - 3*a^2*B + 16*b^2*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)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a + 2*b)*(6*A*b - 3*a*B + 8*b*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)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*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)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)","A",9,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
597,1,562,0,2.0783666,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 \left(49 a^2 A+72 a b B+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{2 \left(88 a^2 A b+75 a^3 B+9 a b^2 B-4 A b^3\right) \sin (c+d x) \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(3 a^2 b (13 A-57 B)+a^3 (-(147 A-75 B))+6 a b^2 (A-3 B)+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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\cos (c+d x)} \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 (9 a B+10 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}","\frac{2 \left(49 a^2 A+72 a b B+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}+\frac{2 \left(88 a^2 A b+75 a^3 B+9 a b^2 B-4 A b^3\right) \sin (c+d x) \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(3 a^2 b (13 A-57 B)+a^3 (-(147 A-75 B))+6 a b^2 (A-3 B)+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(33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right) \sqrt{\cos (c+d x)} \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 (9 a B+10 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}+\frac{2 a 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]*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Sqrt[Cos[c + d*x]]*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 - a^3*(147*A - 75*B) + 3*a^2*b*(13*A - 57*B) + 6*a*b^2*(A - 3*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)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,35,0.1714,1,"{2961, 2989, 3055, 2998, 2816, 2994}"
598,1,473,0,1.5290753,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+42 a b B+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(a^2 (-(25 A-63 B))+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(82 a^2 A b+63 a^3 B+21 a b^2 B-6 A b^3\right) \sqrt{\cos (c+d x)} \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 (7 a B+8 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 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}","\frac{2 \left(25 a^2 A+42 a b B+3 A b^2\right) \sin (c+d x) \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(a^2 (-(25 A-63 B))+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(82 a^2 A b+63 a^3 B+21 a b^2 B-6 A b^3\right) \sqrt{\cos (c+d x)} \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 (7 a B+8 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 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]*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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)])/(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) + 3*a*b*(19*A - 7*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)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (2*(8*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*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,35,0.1714,1,"{2961, 2989, 3055, 2998, 2816, 2994}"
599,1,393,0,1.0905116,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+20 a b B+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)}{15 a^2 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) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{2 (a-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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}} F\left(\sin ^{-1}\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 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} \left(9 a^2 A+20 a b B+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)}{15 a^2 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) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{2 (a-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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}} F\left(\sin ^{-1}\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 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]*(9*a^2*A + 3*A*b^2 + 20*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)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A - 3*A*b - 5*a*B + 15*b*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)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (2*(6*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*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,35,0.1714,1,"{2961, 2989, 3055, 2998, 2816, 2994}"
600,1,479,0,1.0759036,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b} \left(a^2 (A-3 B)-a (4 A b-6 b B)+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}} F\left(\sin ^{-1}\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) \sqrt{a+b} (3 a B+4 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 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b 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}} \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 (A-3 B)-a (4 A b-6 b B)+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}} F\left(\sin ^{-1}\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) \sqrt{a+b} (3 a B+4 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 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b 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}} \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]*(4*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*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(3*A*b^2 + a^2*(A - 3*B) - a*(4*A*b - 6*b*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)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*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)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,35,0.2000,1,"{2961, 2989, 3053, 2809, 2998, 2816, 2994}"
601,1,509,0,1.3800447,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","-\frac{(2 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a (A-B)-b (4 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{(a-b) \sqrt{a+b} (2 a A-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)}{a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (3 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}","-\frac{(2 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a (A-B)-b (4 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{(a-b) \sqrt{a+b} (2 a A-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)}{a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (3 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}",1,"((a - b)*Sqrt[a + b]*(2*a*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*(A - B) - b*(4*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 + 3*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)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,35,0.2286,1,"{2961, 2989, 3061, 3053, 2809, 2998, 2816, 2994}"
602,1,532,0,1.3670736,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \left(3 a^2 B+12 a A b+4 b^2 B\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{(5 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (8 a A+5 a B+4 A b+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}} F\left(\sin ^{-1}\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} (5 a B+4 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 a d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \left(3 a^2 B+12 a A b+4 b^2 B\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{(5 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (8 a A+5 a B+4 A b+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}} F\left(\sin ^{-1}\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} (5 a B+4 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 a d \sqrt{\sec (c+d x)}}+\frac{b B \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*A*b + 5*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)])/(4*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a*A + 4*A*b + 5*a*B + 2*b*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)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(12*a*A*b + 3*a^2*B + 4*b^2*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)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",8,8,35,0.2286,1,"{2961, 2990, 3061, 3053, 2809, 2998, 2816, 2994}"
603,1,626,0,1.9734686,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \left(3 a^2 B+30 a A b+14 a b B+12 A b^2+16 b^2 B\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 B+30 a A b+16 b^2 B\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{\sqrt{a+b} \left(6 a^2 A b+a^3 (-B)+12 a b^2 B+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}} \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{(7 a B+6 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \left(3 a^2 B+30 a A b+14 a b B+12 A b^2+16 b^2 B\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 B+30 a A b+16 b^2 B\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{\sqrt{a+b} \left(6 a^2 A b+a^3 (-B)+12 a b^2 B+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}} \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{(7 a B+6 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"-((a - b)*Sqrt[a + b]*(30*a*A*b + 3*a^2*B + 16*b^2*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)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(30*a*A*b + 12*A*b^2 + 3*a^2*B + 14*a*b*B + 16*b^2*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)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*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)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + ((6*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Sec[c + d*x]]) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)","A",9,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
604,1,730,0,2.4407622,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^2 d}+\frac{\left(-3 a^2 B+8 a A b+12 b^2 B\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(-6 a^2 b (4 A+B)+9 a^3 B-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 B)\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^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 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}} E\left(\sin ^{-1}\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} \left(8 a^3 A b-24 a^2 b^2 B-3 a^4 B-96 a A b^3-48 b^4 B\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{(8 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}","\frac{\left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^2 d}+\frac{\left(-3 a^2 B+8 a A b+12 b^2 B\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(-6 a^2 b (4 A+B)+9 a^3 B-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 B)\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^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(24 a^2 A b-9 a^3 B+156 a b^2 B+128 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}} E\left(\sin ^{-1}\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} \left(8 a^3 A b-24 a^2 b^2 B-3 a^4 B-96 a A b^3-48 b^4 B\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{(8 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{B \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*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*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)])/(192*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(9*a^3*B - 6*a^2*b*(4*A + B) - 8*b^3*(16*A + 9*B) - 4*a*b^2*(28*A + 39*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)])/(192*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*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)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) + ((8*a*A*b - 3*a^2*B + 12*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) + ((8*A*b - 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d)","A",10,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
605,1,662,0,2.9109825,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2),x]","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 d}+\frac{2 \left(1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d}+\frac{2 \left(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^2 b^2 (19 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{\cos (c+d x)} \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)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{11 d}","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 d}+\frac{2 \left(1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d}+\frac{2 \left(1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^2 b^2 (19 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right) \sqrt{\cos (c+d x)} \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)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{11 d}",1,"(2*(a - b)*Sqrt[a + b]*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*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 + 3*a^4*(225*A - 539*B) - 6*a^3*b*(505*A - 209*B) + 15*a^2*b^2*(19*A - 121*B) + 10*a*b^3*(3*A - 11*B))*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*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)","A",9,7,35,0.2000,1,"{2961, 2989, 3047, 3055, 2998, 2816, 2994}"
606,1,562,0,2.0715622,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 \left(49 a^2 A+135 a b B+75 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 \left(163 a^2 A b+75 a^3 B+135 a b^2 B+5 A b^3\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-60 B)+3 a^3 (49 A-25 B)+15 a b^2 (11 A-3 B)+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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \sqrt{\cos (c+d x)} \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 (3 a B+4 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 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}","\frac{2 \left(49 a^2 A+135 a b B+75 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 \left(163 a^2 A b+75 a^3 B+135 a b^2 B+5 A b^3\right) \sin (c+d x) \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-60 B)+3 a^3 (49 A-25 B)+15 a b^2 (11 A-3 B)+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(279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right) \sqrt{\cos (c+d x)} \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 (3 a B+4 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 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]*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*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 - 6*a^2*b*(19*A - 60*B) + 3*a^3*(49*A - 25*B) + 15*a*b^2*(11*A - 3*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)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*a*(4*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*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,7,35,0.2000,1,"{2961, 2989, 3047, 3055, 2998, 2816, 2994}"
607,1,474,0,1.5008545,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 B)\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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 A b+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\cos (c+d x)} \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^2 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 a B+10 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 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 B)\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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(145 a^2 A b+63 a^3 B+161 a b^2 B+15 A b^3\right) \sqrt{\cos (c+d x)} \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^2 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 a B+10 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 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]*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*A - 63*B) + 15*b^2*(A - 7*B) - 8*a*b*(15*A - 7*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)])/(105*a*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(10*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*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,7,35,0.2000,1,"{2961, 2989, 3047, 3055, 2998, 2816, 2994}"
608,1,553,0,1.4727426,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{a+b} \left(a^2 b (17 A-35 B)+a^3 (-(9 A-5 B))-a b^2 (23 A-45 B)+15 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{2 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 a B+8 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 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^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}} \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 b (17 A-35 B)+a^3 (-(9 A-5 B))-a b^2 (23 A-45 B)+15 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{2 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 a B+8 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 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^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}} \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]*(9*a^2*A + 23*A*b^2 + 35*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)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(15*A*b^3 - a*b^2*(23*A - 45*B) + a^2*b*(17*A - 35*B) - a^3*(9*A - 5*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)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*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)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*(8*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*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,8,35,0.2286,1,"{2961, 2989, 3047, 3053, 2809, 2998, 2816, 2994}"
609,1,596,0,1.8954947,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","-\frac{\left(6 a^2 B+14 a A b-3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{\sqrt{a+b} \left(-2 a^2 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-3 b^2 B\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 d \sqrt{\sec (c+d x)}}+\frac{2 a (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{b \sqrt{a+b} (5 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}","-\frac{\left(6 a^2 B+14 a A b-3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{\sqrt{a+b} \left(-2 a^2 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-3 b^2 B\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 d \sqrt{\sec (c+d x)}}+\frac{2 a (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{b \sqrt{a+b} (5 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((a - b)*Sqrt[a + b]*(14*a*A*b + 6*a^2*B - 3*b^2*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*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a*b*(7*A - 9*B) - 2*a^2*(A - 3*B) - 3*b^2*(6*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)])/(3*d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[a + b]*(2*A*b + 5*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)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*(2*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,35,0.2571,1,"{2961, 2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
610,1,607,0,1.8792802,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","-\frac{\left(8 a^2 A-9 a b B-4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \left(8 a^2 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 B+20 a A b+4 b^2 B\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 (4 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}","-\frac{\left(8 a^2 A-9 a b B-4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \left(8 a^2 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 B+20 a A b+4 b^2 B\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 (4 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a 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^2*A - 4*A*b^2 - 9*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)])/(4*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a^2*(A - B) - 2*b^2*(2*A + B) - 3*a*b*(8*A + 3*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)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(20*a*A*b + 15*a^2*B + 4*b^2*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)])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(4*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",9,9,35,0.2571,1,"{2961, 2989, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
611,1,624,0,1.9460634,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(a^2 (48 A+33 B)+a (54 A b+26 b B)+4 b^2 (3 A+4 B)\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(33 a^2 B+54 a A b+16 b^2 B\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{\sqrt{a+b} \left(30 a^2 A b+5 a^3 B+20 a b^2 B+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}} \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 (3 a B+2 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(a^2 (48 A+33 B)+a (54 A b+26 b B)+4 b^2 (3 A+4 B)\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(33 a^2 B+54 a A b+16 b^2 B\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{\sqrt{a+b} \left(30 a^2 A b+5 a^3 B+20 a b^2 B+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}} \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 (3 a B+2 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{b B \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]*(54*a*A*b + 33*a^2*B + 16*b^2*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)])/(24*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(4*b^2*(3*A + 4*B) + a^2*(48*A + 33*B) + a*(54*A*b + 26*b*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)])/(24*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*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)])/(8*b*d*Sqrt[Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d)","A",9,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
612,1,724,0,2.5175281,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\left(5 a^2 B+24 a A b+12 b^2 B\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 (132 A+59 B)+15 a^3 B+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 B)\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(264 a^2 A b+15 a^3 B+284 a b^2 B+128 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}} E\left(\sin ^{-1}\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} \left(40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right) \sqrt{\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{(11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\left(264 a^2 A b+15 a^3 B+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\left(5 a^2 B+24 a A b+12 b^2 B\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 (132 A+59 B)+15 a^3 B+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 B)\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(264 a^2 A b+15 a^3 B+284 a b^2 B+128 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}} E\left(\sin ^{-1}\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} \left(40 a^3 A b+120 a^2 b^2 B-5 a^4 B+160 a A b^3+48 b^4 B\right) \sqrt{\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{(11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"-((a - b)*Sqrt[a + b]*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*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)])/(192*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*B + 8*b^3*(16*A + 9*B) + 2*a^2*b*(132*A + 59*B) + 4*a*b^2*(52*A + 71*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)])/(192*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[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]]) + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + ((24*a*A*b + 5*a^2*B + 12*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + ((8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)","A",10,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
613,1,839,0,3.6027969,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 B a^2+50 A b a+64 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\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 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\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 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\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 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\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{B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 B a^2+50 A b a+64 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\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 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\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 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\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 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\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*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*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)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*B - 30*a^3*b*(5*A + B) - 16*b^4*(45*A + 64*B) - 8*a*b^3*(355*A + 193*B) - 4*a^2*b^2*(295*A + 423*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)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*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)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a^2*A*b + 120*A*b^3 - 15*a^3*B + 172*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) + ((50*a*A*b - 15*a^2*B + 64*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) + ((10*A*b - 3*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)","A",11,9,35,0.2571,1,"{2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
614,1,403,0,1.0256609,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \sqrt{a+b} \left(a^2 (9 A-5 B)-2 a b (A+5 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(9 a^2 A-10 a b 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}} E\left(\sin ^{-1}\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} \left(a^2 (9 A-5 B)-2 a b (A+5 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(9 a^2 A-10 a b 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}} E\left(\sin ^{-1}\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]*(9*a^2*A + 8*A*b^2 - 10*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)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(8*A*b^2 + a^2*(9*A - 5*B) - 2*a*b*(A + 5*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)])/(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",6,6,35,0.1714,1,"{2961, 3000, 3055, 2998, 2816, 2994}"
615,1,330,0,0.6639517,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b} (a (A-3 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)}{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} (a (A-3 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)}{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))*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",5,5,35,0.1429,1,"{2961, 3000, 2998, 2816, 2994}"
616,1,270,0,0.4566111,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*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 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)}}",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]])","A",4,4,35,0.1143,1,"{2961, 2998, 2816, 2994}"
617,1,268,0,0.3992859,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],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 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}} \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 \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 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}} \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*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]*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]])","A",4,4,35,0.1143,1,"{2961, 3006, 2809, 2816}"
618,1,487,0,1.2653578,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{\sqrt{a+b} (2 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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} (2 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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b 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*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*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)])/(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^2*d*Sqrt[Sec[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,35,0.2286,1,"{2961, 3003, 3051, 2809, 2993, 2998, 2816, 2994}"
619,1,539,0,1.2529153,"\int \frac{A+B \cos (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])/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{a+b} \left(-3 a^2 B+4 a A b-4 b^2 B\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{(4 A b-3 a B) \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 B+4 A b+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}} F\left(\sin ^{-1}\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 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)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{B \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 B+4 a A b-4 b^2 B\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{(4 A b-3 a B) \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 B+4 A b+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}} F\left(\sin ^{-1}\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 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)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{B \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*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)])/(4*a*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(4*A*b - 3*a*B + 2*b*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)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(4*a*A*b - 3*a^2*B - 4*b^2*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)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) + ((4*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)","A",8,8,35,0.2286,1,"{2961, 2990, 3061, 3053, 2809, 2998, 2816, 2994}"
620,1,433,0,1.1594334,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a^2 A+3 a b B-4 A b^2\right) \sin (c+d x) \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 b (A b-a B) \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(5 a^2 A b-3 a^3 B+6 a b^2 B-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}} E\left(\sin ^{-1}\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 (a+2 b) (a (A-3 B)+4 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^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2 A+3 a b B-4 A b^2\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 b (A b-a B) \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(5 a^2 A b-3 a^3 B+6 a b^2 B-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}} E\left(\sin ^{-1}\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 (a+2 b) (a (A-3 B)+4 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^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*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^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(a + 2*b)*(4*A*b + a*(A - 3*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)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*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",6,6,35,0.1714,1,"{2961, 3000, 3055, 2998, 2816, 2994}"
621,1,345,0,0.7905471,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 A+a b 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}} E\left(\sin ^{-1}\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-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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}","\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 A+a b 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}} E\left(\sin ^{-1}\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-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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(2*(a^2*A - 2*A*b^2 + 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^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b + a*(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^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 3000, 2998, 2816, 2994}"
622,1,324,0,0.6849799,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{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{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{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{a+b} \sqrt{\sec (c+d x)}}",1,"(2*(A*b - 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*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (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)])/(a*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,35,0.1429,1,"{2961, 2993, 2998, 2816, 2994}"
623,1,476,0,0.7752664,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{2 a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 (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)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{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}} \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 a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 (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)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{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}} \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 - 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*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(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*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*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^2*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,35,0.2000,1,"{2961, 2992, 2809, 2794, 2795, 2816, 2994}"
624,1,560,0,1.5090193,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","-\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}+\frac{2 a (A b-a B) \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 B+2 a A b+b^2 B\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{(2 A b-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}} F\left(\sin ^{-1}\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} \sqrt{\sec (c+d x)}}-\frac{\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}} \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 B+2 a A b+b^2 B\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 a (A b-a B) \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 B+2 a A b+b^2 B\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{(2 A b-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}} F\left(\sin ^{-1}\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} \sqrt{\sec (c+d x)}}-\frac{\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}} \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*A*b - 3*a^2*B + b^2*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*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((2*A*b - (3*a + b)*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)])/(b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b - 3*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^3*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",8,8,35,0.2286,1,"{2961, 2989, 3061, 3053, 2809, 2998, 2816, 2994}"
625,1,607,0,2.2031801,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sin (c+d x) \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{2 b \left(10 a^2 A b-7 a^3 B+3 a b^2 B-6 A b^3\right) \sin (c+d x) \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 b (A b-a B) \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+3 B)-9 a^3 b (A-B)+a^4 (-(A-3 B))+4 a b^3 (3 A-2 B)+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{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\cos (c+d x)} \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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 \left(-13 a^2 A b^2+a^4 A+8 a^3 b B-4 a b^3 B+8 A b^4\right) \sin (c+d x) \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{2 b \left(10 a^2 A b-7 a^3 B+3 a b^2 B-6 A b^3\right) \sin (c+d x) \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 b (A b-a B) \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+3 B)-9 a^3 b (A-B)+a^4 (-(A-3 B))+4 a b^3 (3 A-2 B)+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{2 \left(-28 a^2 A b^3+8 a^4 A b+15 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right) \sqrt{\cos (c+d x)} \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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(16*A*b^4 - a^4*(A - 3*B) + 4*a*b^3*(3*A - 2*B) - 9*a^3*b*(A - B) - 2*a^2*b^2*(8*A + 3*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)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*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*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*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*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",7,6,35,0.1714,1,"{2961, 3000, 3055, 2998, 2816, 2994}"
626,1,496,0,1.3825426,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b \left(8 a^2 A b-5 a^3 B+a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 A+B)-3 a^3 (A-B)+2 a b^2 (3 A-B)+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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\cos (c+d x)} \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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 b \left(8 a^2 A b-5 a^3 B+a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^2 b (3 A+B)-3 a^3 (A-B)+2 a b^2 (3 A-B)+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(-15 a^2 A b^2+3 a^4 A+6 a^3 b B-2 a b^3 B+8 A b^4\right) \sqrt{\cos (c+d x)} \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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[Cos[c + d*x]]*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 - 3*a^3*(A - B) + 2*a*b^2*(3*A - B) - 3*a^2*b*(3*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)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 3000, 3055, 2998, 2816, 2994}"
627,1,469,0,1.1900699,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \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{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(6 a^2 A b-3 a^3 B-a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \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,"(2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b^2 - 3*a^2*(A + B) + a*b*(3*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)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*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*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*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",6,6,35,0.1714,1,"{2961, 3000, 2993, 2998, 2816, 2994}"
628,1,431,0,1.0809364,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{2 \left(3 a^2 A-4 a b B+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2 A-4 a b B+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)}{3 a^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 (a (3 A+B)-b (A+3 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 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2 A-4 a b B+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2 A-4 a b B+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)}{3 a^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 (a (3 A+B)-b (A+3 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 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-2*(3*a^2*A + A*b^2 - 4*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)])/(3*a^2*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*(a*(3*A + B) - b*(A + 3*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)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,35,0.1714,1,"{2961, 2999, 2993, 2998, 2816, 2994}"
629,1,602,0,1.645391,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 a \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 B+3 a^3 B-a b^2 (A+6 B)+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)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sqrt{\cos (c+d x)} \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^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{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}} \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 a \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 B+3 a^3 B-a b^2 (A+6 B)+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)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sqrt{\cos (c+d x)} \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^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{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}} \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*(4*A*b^3 + 3*a^3*B - 7*a*b^2*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*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(3*A*b^3 + 3*a^3*B + a^2*b*B - a*b^2*(A + 6*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)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*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^3*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*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*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,35,0.2286,1,"{2961, 2989, 3051, 2809, 2993, 2998, 2816, 2994}"
630,1,733,0,2.4936351,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{2 a (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{2 a \left(2 a^2 A b-5 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\left(-a^2 (6 A b-5 b B)+15 a^3 B-a b^2 (2 A+21 B)+3 b^3 (4 A-B)\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 b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sqrt{\cos (c+d x)} \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{\sqrt{a+b} (2 A b-5 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 a (A b-a B) \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(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{2 a \left(2 a^2 A b-5 a^3 B+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\left(-a^2 (6 A b-5 b B)+15 a^3 B-a b^2 (2 A+21 B)+3 b^3 (4 A-B)\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 b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left(6 a^3 A b+26 a^2 b^2 B-15 a^4 B-14 a A b^3-3 b^4 B\right) \sqrt{\cos (c+d x)} \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{\sqrt{a+b} (2 A b-5 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,"((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[Cos[c + d*x]]*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]]) + ((3*b^3*(4*A - B) + 15*a^3*B - a*b^2*(2*A + 21*B) - a^2*(6*A*b - 5*b*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)])/(3*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b - 5*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^4*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*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*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)","A",9,9,35,0.2571,1,"{2961, 2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
631,1,266,0,0.3701492,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 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^2 d \sqrt{\sec (c+d x)}}-\frac{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)}{a d \sqrt{\sec (c+d x)}}","\frac{2 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^2 d \sqrt{\sec (c+d x)}}-\frac{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)}{a d \sqrt{\sec (c+d x)}}",1,"(2*(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^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*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]])","A",5,5,38,0.1316,1,"{21, 4222, 2801, 2816, 2994}"
632,1,130,0,0.1550442,"\int \frac{(a B+b B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[((a*B + b*B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{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)}{a d \sqrt{\sec (c+d x)}}","\frac{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)}{a d \sqrt{\sec (c+d x)}}",1,"(2*Sqrt[a + b]*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]])","A",3,3,38,0.07895,1,"{21, 4222, 2816}"
633,1,137,0,0.1550476,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","-\frac{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}} \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 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}} \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*Sqrt[a + b]*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]])","A",3,3,38,0.07895,1,"{21, 4222, 2809}"
634,1,479,0,0.9223621,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b d \sqrt{\sec (c+d x)}}","\frac{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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b 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*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*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)])/(b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*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^2*d*Sqrt[Sec[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,38,0.2632,1,"{21, 4222, 2820, 2809, 3003, 2993, 12, 2801, 2816, 2994}"
635,0,0,0,0.1806304,"\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left((A+B \cos (e+f x)) (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n,x\right)",0,"(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][((a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]))/(c*Cos[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
636,1,644,0,2.0423643,"\int (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","-\frac{c^6 \sin (e+f x) \left(4 a^3 A b \left(m^2-8 m+15\right)+6 a^2 b^2 B \left(m^2-7 m+10\right)+a^4 B \left(m^2-8 m+15\right)+4 a A b^3 \left(m^2-7 m+10\right)+b^4 B \left(m^2-6 m+8\right)\right) (c \sec (e+f x))^{m-6} \, _2F_1\left(\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^5 \sin (e+f x) \left(6 a^2 A b^2 \left(m^2-5 m+4\right)+a^4 A \left(m^2-6 m+8\right)+4 a^3 b B \left(m^2-5 m+4\right)+4 a b^3 B \left(m^2-4 m+3\right)+A b^4 \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \tan (e+f x) \left(4 a^2 A b \left(m^2-4 m+3\right)+a^3 B \left(m^2-4 m+3\right)+a b^2 B \left(5 m^2-13 m+8\right)+2 A b^3 \left(m^2-2 m+4\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (4-m)}-\frac{a^2 c^5 \tan (e+f x) \sec (e+f x) \left(a^2 A (2-m)^2+2 a b B (1-m)^2+A b^2 \left(m^2-m+6\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (3-m)}-\frac{a c^5 \tan (e+f x) (a B (1-m)-A b (m+2)) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-5}}{f (1-m) (2-m)}-\frac{a A c^5 \tan (e+f x) (a \sec (e+f x)+b)^3 (c \sec (e+f x))^{m-5}}{f (1-m)}","-\frac{c^6 \sin (e+f x) \left(4 a^3 A b \left(m^2-8 m+15\right)+6 a^2 b^2 B \left(m^2-7 m+10\right)+a^4 B \left(m^2-8 m+15\right)+4 a A b^3 \left(m^2-7 m+10\right)+b^4 B \left(m^2-6 m+8\right)\right) (c \sec (e+f x))^{m-6} \, _2F_1\left(\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^5 \sin (e+f x) \left(6 a^2 A b^2 \left(m^2-5 m+4\right)+a^4 A \left(m^2-6 m+8\right)+4 a^3 b B \left(m^2-5 m+4\right)+4 a b^3 B \left(m^2-4 m+3\right)+A b^4 \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \tan (e+f x) \left(4 a^2 A b \left(m^2-4 m+3\right)+a^3 B \left(m^2-4 m+3\right)+a b^2 B \left(5 m^2-13 m+8\right)+2 A b^3 \left(m^2-2 m+4\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (4-m)}-\frac{a^2 c^5 \tan (e+f x) \sec (e+f x) \left(a^2 A (2-m)^2+2 a b B (1-m)^2+A b^2 \left(m^2-m+6\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (3-m)}-\frac{a c^5 \tan (e+f x) (a B (1-m)-A b (m+2)) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-5}}{f (1-m) (2-m)}-\frac{a A c^5 \tan (e+f x) (a \sec (e+f x)+b)^3 (c \sec (e+f x))^{m-5}}{f (1-m)}",1,"-((c^6*(4*a^3*A*b*(15 - 8*m + m^2) + a^4*B*(15 - 8*m + m^2) + 4*a*A*b^3*(10 - 7*m + m^2) + 6*a^2*b^2*B*(10 - 7*m + m^2) + b^4*B*(8 - 6*m + m^2))*Hypergeometric2F1[1/2, (6 - m)/2, (8 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-6 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*(6 - m)*Sqrt[Sin[e + f*x]^2])) - (c^5*(a^4*A*(8 - 6*m + m^2) + 6*a^2*A*b^2*(4 - 5*m + m^2) + 4*a^3*b*B*(4 - 5*m + m^2) + A*b^4*(3 - 4*m + m^2) + 4*a*b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^5*(4*a^2*A*b*(3 - 4*m + m^2) + a^3*B*(3 - 4*m + m^2) + 2*A*b^3*(4 - 2*m + m^2) + a*b^2*B*(8 - 13*m + 5*m^2))*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(4 - m)) - (a^2*c^5*(2*a*b*B*(1 - m)^2 + a^2*A*(2 - m)^2 + A*b^2*(6 - m + m^2))*Sec[e + f*x]*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(3 - m)) - (a*c^5*(a*B*(1 - m) - A*b*(2 + m))*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^5*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(1 - m))","A",10,8,33,0.2424,1,"{2960, 4026, 4096, 4076, 4047, 3772, 2643, 4046}"
637,1,455,0,1.1454781,"\int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","-\frac{c^5 \sin (e+f x) \left(a^3 A \left(m^2-6 m+8\right)+3 a^2 b B \left(m^2-5 m+4\right)+3 a A b^2 \left(m^2-5 m+4\right)+b^3 B \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^4 \sin (e+f x) \left(3 a^2 A b (3-m)+a^3 B (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^4 \tan (e+f x) \left(a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right) (c \sec (e+f x))^{m-4}}{f (1-m) (3-m)}-\frac{a^2 c^4 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (1-m) (2-m)}-\frac{a A c^4 \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}","-\frac{c^5 \sin (e+f x) \left(a^3 A \left(m^2-6 m+8\right)+3 a^2 b B \left(m^2-5 m+4\right)+3 a A b^2 \left(m^2-5 m+4\right)+b^3 B \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^4 \sin (e+f x) \left(3 a^2 A b (3-m)+a^3 B (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^4 \tan (e+f x) \left(a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right) (c \sec (e+f x))^{m-4}}{f (1-m) (3-m)}-\frac{a^2 c^4 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (1-m) (2-m)}-\frac{a A c^4 \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}",1,"-((c^5*(a^3*A*(8 - 6*m + m^2) + 3*a*A*b^2*(4 - 5*m + m^2) + 3*a^2*b*B*(4 - 5*m + m^2) + b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2])) - (c^4*(A*b^3*(2 - m) + 3*a*b^2*B*(2 - m) + 3*a^2*A*b*(3 - m) + a^3*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^4*(3*a*b*B*(1 - m) + a^2*A*(2 - m) - 2*A*b^2*m)*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(3 - m)) - (a^2*c^4*(a*B*(1 - m) - A*b*(1 + m))*Sec[e + f*x]*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^4*(c*Sec[e + f*x])^(-4 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m))","A",9,7,33,0.2121,1,"{2960, 4026, 4076, 4047, 3772, 2643, 4046}"
638,1,327,0,0.6427959,"\int (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","-\frac{c^4 \sin (e+f x) \left(a^2 B (3-m)+2 a A b (3-m)+b^2 B (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^3 \sin (e+f x) \left(a^2 A (2-m)+2 a b B (1-m)+A b^2 (1-m)\right) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^3 \tan (e+f x) (a B (1-m)-A b m) (c \sec (e+f x))^{m-3}}{f (1-m) (2-m)}-\frac{a A c^3 \tan (e+f x) (a \sec (e+f x)+b) (c \sec (e+f x))^{m-3}}{f (1-m)}","-\frac{c^4 \sin (e+f x) \left(a^2 B (3-m)+2 a A b (3-m)+b^2 B (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^3 \sin (e+f x) \left(a^2 A (2-m)+2 a b B (1-m)+A b^2 (1-m)\right) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^3 \tan (e+f x) (a B (1-m)-A b m) (c \sec (e+f x))^{m-3}}{f (1-m) (2-m)}-\frac{a A c^3 \tan (e+f x) (a \sec (e+f x)+b) (c \sec (e+f x))^{m-3}}{f (1-m)}",1,"-((c^4*(b^2*B*(2 - m) + 2*a*A*b*(3 - m) + a^2*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*Sqrt[Sin[e + f*x]^2])) - (c^3*(A*b^2*(1 - m) + 2*a*b*B*(1 - m) + a^2*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^3*(a*B*(1 - m) - A*b*m)*(c*Sec[e + f*x])^(-3 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^3*(c*Sec[e + f*x])^(-3 + m)*(b + a*Sec[e + f*x])*Tan[e + f*x])/(f*(1 - m))","A",8,6,33,0.1818,1,"{2960, 4026, 4047, 3772, 2643, 4046}"
639,1,217,0,0.3596615,"\int (a+b \cos (e+f x)) (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","-\frac{c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(e+f x)\right)}{f (2-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)}","-\frac{c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(e+f x)\right)}{f (2-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)}",1,"-((c^3*(b*B*(1 - m) + a*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*Sqrt[Sin[e + f*x]^2])) - ((A*b + a*B)*c^2*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-2 + m)*Sin[e + f*x])/(f*(2 - m)*Sqrt[Sin[e + f*x]^2]) - (a*A*c^2*(c*Sec[e + f*x])^(-2 + m)*Tan[e + f*x])/(f*(1 - m))","A",7,5,31,0.1613,1,"{2960, 3997, 3787, 3772, 2643}"
640,1,299,0,0.5758938,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{a+b \cos (e+f x)} \, dx","Int[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/(a + b*Cos[e + f*x]),x]","-\frac{(A b-a B) \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{c f \left(a^2-b^2\right)}+\frac{a (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{m+1}{2}} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m+1}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b c f \left(a^2-b^2\right)}-\frac{B c \sin (e+f x) (c \sec (e+f x))^{m-1} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(e+f x)\right)}{b f (1-m) \sqrt{\sin ^2(e+f x)}}","-\frac{(A b-a B) \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{c f \left(a^2-b^2\right)}+\frac{a (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{m+1}{2}} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m+1}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b c f \left(a^2-b^2\right)}-\frac{B c \sin (e+f x) (c \sec (e+f x))^{m-1} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(e+f x)\right)}{b f (1-m) \sqrt{\sin ^2(e+f x)}}",1,"-(((A*b - a*B)*AppellF1[1/2, m/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(Cos[e + f*x]^2)^(m/2)*(c*Sec[e + f*x])^(1 + m)*Sin[e + f*x])/((a^2 - b^2)*c*f)) + (a*(A*b - a*B)*AppellF1[1/2, (1 + m)/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(Cos[e + f*x]^2)^((1 + m)/2)*(c*Sec[e + f*x])^(1 + m)*Sin[e + f*x])/(b*(a^2 - b^2)*c*f) - (B*c*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-1 + m)*Sin[e + f*x])/(b*f*(1 - m)*Sqrt[Sin[e + f*x]^2])","A",10,8,33,0.2424,1,"{2960, 4038, 3772, 2643, 3869, 2823, 3189, 429}"
641,0,0,0,0.7211171,"\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","\frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(c \cos (e+f x))^{-m} \left(\frac{1}{2} c \cos (e+f x) \left(a (5-2 m) (a B+2 A b)+b^2 B (3-2 m)\right)+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (3-m)+A b (5-2 m))+\frac{1}{2} a c \left(2 a A \left(\frac{5}{2}-m\right)+2 b B (1-m)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{c (5-2 m)}+\frac{2 b B \sin (e+f x) \cos (e+f x) \sqrt{a+b \cos (e+f x)} (c \sec (e+f x))^m}{f (5-2 m)}",0,"(2*b*B*Cos[e + f*x]*Sqrt[a + b*Cos[e + f*x]]*(c*Sec[e + f*x])^m*Sin[e + f*x])/(f*(5 - 2*m)) + (2*(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][((a*c*(2*b*B*(1 - m) + 2*a*A*(5/2 - m)))/2 + (c*(b^2*B*(3 - 2*m) + a*(2*A*b + a*B)*(5 - 2*m))*Cos[e + f*x])/2 + (b*c*(A*b*(5 - 2*m) + 2*a*B*(3 - m))*Cos[e + f*x]^2)/2)/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x])/(c*(5 - 2*m))","A",0,0,0,0,-1,"{}"
642,0,0,0,0.2387089,"\int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Int[Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \cos (e+f x))^{-m},x\right)",0,"(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][(Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]))/(c*Cos[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
643,0,0,0,0.2416765,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx","Int[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/Sqrt[a + b*Cos[e + f*x]],x]","\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^{-m}}{\sqrt{a+b \cos (e+f x)}},x\right)",0,"(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][(A + B*Cos[e + f*x])/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x]","A",0,0,0,0,-1,"{}"
644,0,0,0,0.6864318,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx","Int[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/(a + b*Cos[e + f*x])^(3/2),x]","\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx","\frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(c \cos (e+f x))^{-m} \left(\frac{1}{2} c \left(a^2 A-2 a b B (1-m)+A b^2 (1-2 m)\right)-\frac{1}{2} b c (3-2 m) (A b-a B) \cos ^2(e+f x)-\frac{1}{2} a c (A b-a B) \cos (e+f x)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{a c \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (e+f x) \cos (e+f x) (c \sec (e+f x))^m}{a f \left(a^2-b^2\right) \sqrt{a+b \cos (e+f x)}}",0,"(2*b*(A*b - a*B)*Cos[e + f*x]*(c*Sec[e + f*x])^m*Sin[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Cos[e + f*x]]) + (2*(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Defer[Int][((c*(a^2*A + A*b^2*(1 - 2*m) - 2*a*b*B*(1 - m)))/2 - (a*(A*b - a*B)*c*Cos[e + f*x])/2 - (b*(A*b - a*B)*c*(3 - 2*m)*Cos[e + f*x]^2)/2)/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x])/(a*(a^2 - b^2)*c)","A",0,0,0,0,-1,"{}"