1,1,114,0,0.0695691,"\int \cos ^5(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Cos[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(5*a*x)/16 + (a*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",7,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
2,1,92,0,0.0575704,"\int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Cos[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(3*a*x)/8 + (a*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",6,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
3,1,76,0,0.0524819,"\int \cos ^3(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(3*a*x)/8 + (a*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*Sin[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
4,1,54,0,0.04155,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a*Sin[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
5,1,38,0,0.0144199,"\int \cos (c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}","\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,17,0.05882,1,"{2734}"
6,1,15,0,0.0072543,"\int (a+a \cos (c+d x)) \, dx","Int[a + a*Cos[c + d*x],x]","\frac{a \sin (c+d x)}{d}+a x","\frac{a \sin (c+d x)}{d}+a x",1,"a*x + (a*Sin[c + d*x])/d","A",2,1,10,0.1000,1,"{2637}"
7,1,16,0,0.0201203,"\int (a+a \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x",1,"a*x + (a*ArcTanh[Sin[c + d*x]])/d","A",2,2,17,0.1176,1,"{2735, 3770}"
8,1,24,0,0.0337479,"\int (a+a \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d","A",4,4,19,0.2105,1,"{2748, 3767, 8, 3770}"
9,1,47,0,0.0475257,"\int (a+a \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,19,0.2632,1,"{2748, 3768, 3770, 3767, 8}"
10,1,63,0,0.0477197,"\int (a+a \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Tan[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{2748, 3767, 3768, 3770}"
11,1,85,0,0.0616641,"\int (a+a \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{2748, 3768, 3770, 3767}"
12,1,101,0,0.0657615,"\int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)","A",6,4,19,0.2105,1,"{2748, 3767, 3768, 3770}"
13,1,129,0,0.1295576,"\int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2,x]","\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{4 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{11 a^2 x}{16}","\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{4 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{11 a^2 x}{16}",1,"(11*a^2*x)/16 + (2*a^2*Sin[c + d*x])/d + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a^2*Sin[c + d*x]^3)/(3*d) + (2*a^2*Sin[c + d*x]^5)/(5*d)","A",11,4,21,0.1905,1,"{2757, 2635, 8, 2633}"
14,1,103,0,0.1042874,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4}","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4}",1,"(3*a^2*x)/4 + (2*a^2*Sin[c + d*x])/d + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - (a^2*Sin[c + d*x]^3)/d + (a^2*Sin[c + d*x]^5)/(5*d)","A",9,4,21,0.1905,1,"{2757, 2633, 2635, 8}"
15,1,87,0,0.0974391,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2,x]","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8}","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8}",1,"(7*a^2*x)/8 + (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a^2*Sin[c + d*x]^3)/(3*d)","A",9,4,21,0.1905,1,"{2757, 2635, 8, 2633}"
16,1,69,0,0.0414491,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2,x]","\frac{4 a^2 \sin (c+d x)}{3 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{3 d}+a^2 x+\frac{\sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}","-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x",1,"a^2*x + (4*a^2*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(3*d) + ((a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",2,2,19,0.1053,1,"{2751, 2644}"
17,1,45,0,0.0140222,"\int (a+a \cos (c+d x))^2 \, dx","Int[(a + a*Cos[c + d*x])^2,x]","\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}","\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}",1,"(3*a^2*x)/2 + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,12,0.08333,1,"{2644}"
18,1,34,0,0.0575523,"\int (a+a \cos (c+d x))^2 \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x],x]","\frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x","\frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x",1,"2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d","A",3,3,19,0.1579,1,"{2746, 2735, 3770}"
19,1,34,0,0.0583988,"\int (a+a \cos (c+d x))^2 \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x",1,"a^2*x + (2*a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d","A",5,4,21,0.1905,1,"{2757, 3770, 3767, 8}"
20,1,54,0,0.0788182,"\int (a+a \cos (c+d x))^2 \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3,x]","\frac{2 a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{2 a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,5,21,0.2381,1,"{2757, 3770, 3767, 8, 3768}"
21,1,66,0,0.0877679,"\int (a+a \cos (c+d x))^2 \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/d + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)","A",8,5,21,0.2381,1,"{2757, 3767, 8, 3768, 3770}"
22,1,96,0,0.1079702,"\int (a+a \cos (c+d x))^2 \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^5,x]","\frac{2 a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{7 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{7 a^2 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{2 a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{7 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{7 a^2 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(7*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*Tan[c + d*x])/d + (7*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a^2*Tan[c + d*x]^3)/(3*d)","A",9,4,21,0.1905,1,"{2757, 3768, 3770, 3767}"
23,1,129,0,0.1465918,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3,x]","\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{7 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{23 a^3 x}{16}","\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{7 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{23 a^3 x}{16}",1,"(23*a^3*x)/16 + (4*a^3*Sin[c + d*x])/d + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (7*a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^5)/(5*d)","A",13,4,21,0.1905,1,"{2757, 2633, 2635, 8}"
24,1,105,0,0.1172395,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3,x]","\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8}","\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8}",1,"(13*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^5)/(5*d)","A",11,4,21,0.1905,1,"{2757, 2635, 8, 2633}"
25,1,88,0,0.0780084,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^3,x]","-\frac{a^3 \sin ^3(c+d x)}{4 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 x}{8}+\frac{\sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}","-\frac{a^3 \sin ^3(c+d x)}{d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 x}{8}",1,"(15*a^3*x)/8 + (3*a^3*Sin[c + d*x])/d + (9*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/(4*d)","A",8,6,19,0.3158,1,"{2751, 2645, 2637, 2635, 8, 2633}"
26,1,63,0,0.0526376,"\int (a+a \cos (c+d x))^3 \, dx","Int[(a + a*Cos[c + d*x])^3,x]","-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}","-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}",1,"(5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)","A",7,5,12,0.4167,1,"{2645, 2637, 2635, 8, 2633}"
27,1,59,0,0.0623902,"\int (a+a \cos (c+d x))^3 \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x],x]","\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2}","\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2}",1,"(7*a^3*x)/2 + (a^3*ArcTanh[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,5,19,0.2632,1,"{2757, 2637, 2635, 8, 3770}"
28,1,48,0,0.0669439,"\int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2,x]","\frac{a^3 \sin (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x","\frac{a^3 \sin (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x",1,"3*a^3*x + (3*a^3*ArcTanh[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d + (a^3*Tan[c + d*x])/d","A",6,5,21,0.2381,1,"{2757, 2637, 3770, 3767, 8}"
29,1,59,0,0.0826457,"\int (a+a \cos (c+d x))^3 \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3,x]","\frac{3 a^3 \tan (c+d x)}{d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}+a^3 x","\frac{3 a^3 \tan (c+d x)}{d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}+a^3 x",1,"a^3*x + (7*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,5,21,0.2381,1,"{2757, 3770, 3767, 8, 3768}"
30,1,72,0,0.0950675,"\int (a+a \cos (c+d x))^3 \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4,x]","\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (4*a^3*Tan[c + d*x])/d + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^3*Tan[c + d*x]^3)/(3*d)","A",9,5,21,0.2381,1,"{2757, 3770, 3767, 8, 3768}"
31,1,93,0,0.1165384,"\int (a+a \cos (c+d x))^3 \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5,x]","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d","A",11,5,21,0.2381,1,"{2757, 3767, 8, 3768, 3770}"
32,1,114,0,0.1274245,"\int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^6,x]","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{5 a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{5 a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (5*a^3*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d)","A",11,4,21,0.1905,1,"{2757, 3768, 3770, 3767}"
33,1,127,0,0.1566393,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4,x]","\frac{4 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{41 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{49 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{49 a^4 x}{16}","\frac{4 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{41 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{49 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{49 a^4 x}{16}",1,"(49*a^4*x)/16 + (8*a^4*Sin[c + d*x])/d + (49*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (41*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^4*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a^4*Sin[c + d*x]^3)/d + (4*a^4*Sin[c + d*x]^5)/(5*d)","A",15,4,21,0.1905,1,"{2757, 2635, 8, 2633}"
34,1,114,0,0.1069148,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]","-\frac{16 a^4 \sin ^3(c+d x)}{15 d}+\frac{32 a^4 \sin (c+d x)}{5 d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{5 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{10 d}+\frac{7 a^4 x}{2}+\frac{\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2}",1,"(7*a^4*x)/2 + (32*a^4*Sin[c + d*x])/(5*d) + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(10*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(5*d) + ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) - (16*a^4*Sin[c + d*x]^3)/(15*d)","A",11,6,19,0.3158,1,"{2751, 2645, 2637, 2635, 8, 2633}"
35,1,87,0,0.0809796,"\int (a+a \cos (c+d x))^4 \, dx","Int[(a + a*Cos[c + d*x])^4,x]","-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}","-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}",1,"(35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)","A",10,5,12,0.4167,1,"{2645, 2637, 2635, 8, 2633}"
36,1,73,0,0.080568,"\int (a+a \cos (c+d x))^4 \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x],x]","-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{7 a^4 \sin (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \sin (c+d x) \cos (c+d x)}{d}+6 a^4 x","-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{7 a^4 \sin (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \sin (c+d x) \cos (c+d x)}{d}+6 a^4 x",1,"6*a^4*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Sin[c + d*x])/d + (2*a^4*Cos[c + d*x]*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/(3*d)","A",8,6,19,0.3158,1,"{2757, 2637, 2635, 8, 2633, 3770}"
37,1,73,0,0.0828627,"\int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2,x]","\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2}","\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2}",1,"(13*a^4*x)/2 + (4*a^4*ArcTanh[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Tan[c + d*x])/d","A",8,6,21,0.2857,1,"{2757, 2637, 2635, 8, 3770, 3767}"
38,1,73,0,0.0871818,"\int (a+a \cos (c+d x))^4 \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3,x]","\frac{a^4 \sin (c+d x)}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x","\frac{a^4 \sin (c+d x)}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x",1,"4*a^4*x + (13*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*Sin[c + d*x])/d + (4*a^4*Tan[c + d*x])/d + (a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",8,6,21,0.2857,1,"{2757, 2637, 3770, 3767, 8, 3768}"
39,1,73,0,0.0956729,"\int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4,x]","\frac{a^4 \tan ^3(c+d x)}{3 d}+\frac{7 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \tan (c+d x) \sec (c+d x)}{d}+a^4 x","\frac{a^4 \tan ^3(c+d x)}{3 d}+\frac{7 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \tan (c+d x) \sec (c+d x)}{d}+a^4 x",1,"a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Tan[c + d*x])/d + (2*a^4*Sec[c + d*x]*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/(3*d)","A",9,5,21,0.2381,1,"{2757, 3770, 3767, 8, 3768}"
40,1,96,0,0.1257085,"\int (a+a \cos (c+d x))^4 \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5,x]","\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(3*d)","A",12,5,21,0.2381,1,"{2757, 3770, 3767, 8, 3768}"
41,1,111,0,0.1436889,"\int (a+a \cos (c+d x))^4 \sec ^6(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6,x]","\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{8 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{d}+\frac{7 a^4 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{8 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{d}+\frac{7 a^4 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(7*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (8*a^4*Tan[c + d*x])/d + (7*a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/d + (8*a^4*Tan[c + d*x]^3)/(3*d) + (a^4*Tan[c + d*x]^5)/(5*d)","A",13,5,21,0.2381,1,"{2757, 3767, 8, 3768, 3770}"
42,1,136,0,0.1822639,"\int (a+a \cos (c+d x))^4 \sec ^7(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^7,x]","\frac{4 a^4 \tan ^5(c+d x)}{5 d}+\frac{4 a^4 \tan ^3(c+d x)}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{49 a^4 \tan (c+d x) \sec (c+d x)}{16 d}","\frac{4 a^4 \tan ^5(c+d x)}{5 d}+\frac{4 a^4 \tan ^3(c+d x)}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{49 a^4 \tan (c+d x) \sec (c+d x)}{16 d}",1,"(49*a^4*ArcTanh[Sin[c + d*x]])/(16*d) + (8*a^4*Tan[c + d*x])/d + (49*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (41*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*Tan[c + d*x]^3)/d + (4*a^4*Tan[c + d*x]^5)/(5*d)","A",15,4,21,0.1905,1,"{2757, 3768, 3770, 3767}"
43,1,118,0,0.1068837,"\int \frac{\cos ^5(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x]),x]","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{15 x}{8 a}","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{15 x}{8 a}",1,"(15*x)/(8*a) - (4*Sin[c + d*x])/(a*d) + (15*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + (4*Sin[c + d*x]^3)/(3*a*d)","A",7,5,21,0.2381,1,"{2767, 2748, 2633, 2635, 8}"
44,1,94,0,0.0923199,"\int \frac{\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x]),x]","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 x}{2 a}","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 x}{2 a}",1,"(-3*x)/(2*a) + (4*Sin[c + d*x])/(a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (4*Sin[c + d*x]^3)/(3*a*d)","A",6,5,21,0.2381,1,"{2767, 2748, 2635, 8, 2633}"
45,1,76,0,0.0611472,"\int \frac{\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x]),x]","-\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x}{2 a}","-\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x}{2 a}",1,"(3*x)/(2*a) - (2*Sin[c + d*x])/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",2,2,21,0.09524,1,"{2767, 2734}"
46,1,43,0,0.0798836,"\int \frac{\cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x]),x]","\frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x}{a}","\frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x}{a}",1,"-(x/a) + Sin[c + d*x]/(a*d) + Sin[c + d*x]/(a*d*(1 + Cos[c + d*x]))","A",4,4,21,0.1905,1,"{2746, 12, 2735, 2648}"
47,1,29,0,0.0339496,"\int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x]),x]","\frac{x}{a}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}","\frac{x}{a}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))","A",2,2,19,0.1053,1,"{2735, 2648}"
48,1,22,0,0.0123108,"\int \frac{1}{a+a \cos (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(-1),x]","\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}","\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))","A",1,1,12,0.08333,1,"{2648}"
49,1,38,0,0.0475856,"\int \frac{\sec (c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"ArcTanh[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))","A",3,3,19,0.1579,1,"{2747, 3770, 2648}"
50,1,53,0,0.0758197,"\int \frac{\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x]),x]","\frac{2 \tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \cos (c+d x)+a)}","\frac{2 \tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"-(ArcTanh[Sin[c + d*x]]/(a*d)) + (2*Tan[c + d*x])/(a*d) - Tan[c + d*x]/(d*(a + a*Cos[c + d*x]))","A",5,5,21,0.2381,1,"{2768, 2748, 3767, 8, 3770}"
51,1,83,0,0.0931044,"\int \frac{\sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x]),x]","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(3*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*Tan[c + d*x])/(a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2768, 2748, 3768, 3770, 3767, 8}"
52,1,103,0,0.0954953,"\int \frac{\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + a*Cos[c + d*x]),x]","\frac{4 \tan ^3(c+d x)}{3 a d}+\frac{4 \tan (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}","\frac{4 \tan ^3(c+d x)}{3 a d}+\frac{4 \tan (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(-3*ArcTanh[Sin[c + d*x]])/(2*a*d) + (4*Tan[c + d*x])/(a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + (4*Tan[c + d*x]^3)/(3*a*d)","A",6,5,21,0.2381,1,"{2768, 2748, 3767, 3768, 3770}"
53,1,124,0,0.1846654,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^2,x]","-\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-5*x)/a^2 + (12*Sin[c + d*x])/(a^2*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (10*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (4*Sin[c + d*x]^3)/(a^2*d)","A",7,6,21,0.2857,1,"{2765, 2977, 2748, 2635, 8, 2633}"
54,1,114,0,0.1524698,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^2,x]","-\frac{16 \sin (c+d x)}{3 a^2 d}-\frac{8 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{16 \sin (c+d x)}{3 a^2 d}-\frac{8 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(7*x)/(2*a^2) - (16*Sin[c + d*x])/(3*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (8*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,21,0.1429,1,"{2765, 2977, 2734}"
55,1,80,0,0.1695677,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^2,x]","\frac{4 \sin (c+d x)}{3 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{4 \sin (c+d x)}{3 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*x)/a^2 + (4*Sin[c + d*x])/(3*a^2*d) + (2*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,21,0.2857,1,"{2765, 2968, 3023, 12, 2735, 2648}"
56,1,57,0,0.0845487,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^2,x]","-\frac{5 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"x/a^2 - (5*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,21,0.1429,1,"{2758, 2735, 2648}"
57,1,55,0,0.0384962,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^2,x]","\frac{2 \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))","A",2,2,19,0.1053,1,"{2750, 2648}"
58,1,55,0,0.0269949,"\int \frac{1}{(a+a \cos (c+d x))^2} \, dx","Int[(a + a*Cos[c + d*x])^(-2),x]","\frac{\sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{\sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(3*d*(a^2 + a^2*Cos[c + d*x]))","A",2,2,12,0.1667,1,"{2650, 2648}"
59,1,66,0,0.1115542,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^2,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{4 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{4 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(a^2*d) - (4*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)","A",4,4,19,0.2105,1,"{2766, 2978, 12, 3770}"
60,1,81,0,0.1728211,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^2,x]","\frac{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*ArcTanh[Sin[c + d*x]])/(a^2*d) + (10*Tan[c + d*x])/(3*a^2*d) - (2*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - Tan[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,21,0.2857,1,"{2766, 2978, 2748, 3767, 8, 3770}"
61,1,119,0,0.1897504,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^2,x]","-\frac{16 \tan (c+d x)}{3 a^2 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{8 \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{16 \tan (c+d x)}{3 a^2 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{8 \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(7*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (16*Tan[c + d*x])/(3*a^2*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (8*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,21,0.3333,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
62,1,133,0,0.1991965,"\int \frac{\sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + a*Cos[c + d*x])^2,x]","\frac{4 \tan ^3(c+d x)}{a^2 d}+\frac{12 \tan (c+d x)}{a^2 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{10 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{4 \tan ^3(c+d x)}{a^2 d}+\frac{12 \tan (c+d x)}{a^2 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{10 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-5*ArcTanh[Sin[c + d*x]])/(a^2*d) + (12*Tan[c + d*x])/(a^2*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (10*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*Tan[c + d*x]^3)/(a^2*d)","A",7,6,21,0.2857,1,"{2766, 2978, 2748, 3767, 3768, 3770}"
63,1,153,0,0.2647659,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3,x]","-\frac{152 \sin (c+d x)}{15 a^3 d}-\frac{76 \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{13 x}{2 a^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{11 \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{152 \sin (c+d x)}{15 a^3 d}-\frac{76 \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{13 x}{2 a^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{11 \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(13*x)/(2*a^3) - (152*Sin[c + d*x])/(15*a^3*d) + (13*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",4,3,21,0.1429,1,"{2765, 2977, 2734}"
64,1,119,0,0.2725698,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3,x]","\frac{9 \sin (c+d x)}{5 a^3 d}+\frac{3 \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{3 \sin (c+d x) \cos ^2(c+d x)}{5 a d (a \cos (c+d x)+a)^2}","\frac{9 \sin (c+d x)}{5 a^3 d}+\frac{3 \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{3 \sin (c+d x) \cos ^2(c+d x)}{5 a d (a \cos (c+d x)+a)^2}",1,"(-3*x)/a^3 + (9*Sin[c + d*x])/(5*a^3*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (3*Cos[c + d*x]^2*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (3*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,7,21,0.3333,1,"{2765, 2977, 2968, 3023, 12, 2735, 2648}"
65,1,96,0,0.183956,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^3,x]","-\frac{29 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{29 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"x/a^3 - (Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (7*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (29*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,5,21,0.2381,1,"{2765, 2968, 3019, 2735, 2648}"
66,1,83,0,0.0937042,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^3,x]","\frac{7 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{7 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (7*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,21,0.1429,1,"{2758, 2750, 2648}"
67,1,83,0,0.0581915,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^3,x]","\frac{\sin (c+d x)}{5 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{\sin (c+d x)}{5 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + Sin[c + d*x]/(5*a*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(5*d*(a^3 + a^3*Cos[c + d*x]))","A",3,3,19,0.1579,1,"{2750, 2650, 2648}"
68,1,83,0,0.0463566,"\int \frac{1}{(a+a \cos (c+d x))^3} \, dx","Int[(a + a*Cos[c + d*x])^(-3),x]","\frac{2 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{2 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",3,2,12,0.1667,1,"{2650, 2648}"
69,1,97,0,0.2011088,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^3,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{22 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{22 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (7*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (22*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",5,4,19,0.2105,1,"{2766, 2978, 12, 3770}"
70,1,112,0,0.2789017,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^3,x]","\frac{24 \tan (c+d x)}{5 a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 \tan (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}","\frac{24 \tan (c+d x)}{5 a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 \tan (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(-3*ArcTanh[Sin[c + d*x]])/(a^3*d) + (24*Tan[c + d*x])/(5*a^3*d) - Tan[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (3*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (3*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,21,0.2857,1,"{2766, 2978, 2748, 3767, 8, 3770}"
71,1,156,0,0.3045363,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^3,x]","-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{76 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{11 \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}","-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{76 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{11 \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(13*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (152*Tan[c + d*x])/(15*a^3*d) + (13*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - (Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,21,0.3333,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
72,1,184,0,0.3818519,"\int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4,x]","-\frac{576 \sin (c+d x)}{35 a^4 d}-\frac{43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{21 x}{2 a^4}-\frac{\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}","-\frac{576 \sin (c+d x)}{35 a^4 d}-\frac{43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{21 x}{2 a^4}-\frac{\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"(21*x)/(2*a^4) - (576*Sin[c + d*x])/(35*a^4*d) + (21*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - (43*Cos[c + d*x]^3*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) - (288*Cos[c + d*x]^2*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",5,3,21,0.1429,1,"{2765, 2977, 2734}"
73,1,150,0,0.3725737,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^4,x]","\frac{244 \sin (c+d x)}{105 a^4 d}-\frac{88 \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 x}{a^4}-\frac{\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{12 \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{244 \sin (c+d x)}{105 a^4 d}-\frac{88 \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 x}{a^4}-\frac{\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{12 \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(-4*x)/a^4 + (244*Sin[c + d*x])/(105*a^4*d) - (88*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (12*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,7,21,0.3333,1,"{2765, 2977, 2968, 3023, 12, 2735, 2648}"
74,1,127,0,0.2825473,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^4,x]","-\frac{43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac{x}{a^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3}","-\frac{43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac{x}{a^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3}",1,"x/a^4 + (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (43*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3)","A",6,6,21,0.2857,1,"{2765, 2977, 2968, 3019, 2735, 2648}"
75,1,114,0,0.1993597,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^4,x]","\frac{12 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{18 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{8 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}","\frac{12 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{18 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{8 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(-18*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) + (12*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (8*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",5,5,21,0.2381,1,"{2765, 2968, 3019, 2750, 2648}"
76,1,112,0,0.1120203,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^4,x]","\frac{13 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{13 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}-\frac{11 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{13 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{13 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}-\frac{11 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (11*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (13*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (13*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",4,4,21,0.1905,1,"{2758, 2750, 2650, 2648}"
77,1,112,0,0.0783698,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^4,x]","\frac{8 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{8 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{8 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{8 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"-Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))","A",4,3,19,0.1579,1,"{2750, 2650, 2648}"
78,1,112,0,0.0697271,"\int \frac{1}{(a+a \cos (c+d x))^4} \, dx","Int[(a + a*Cos[c + d*x])^(-4),x]","\frac{2 \sin (c+d x)}{35 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 \sin (c+d x)}{35 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{3 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{2 \sin (c+d x)}{35 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 \sin (c+d x)}{35 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{3 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) + (3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(35*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(35*d*(a^4 + a^4*Cos[c + d*x]))","A",4,2,12,0.1667,1,"{2650, 2648}"
79,1,120,0,0.2869679,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^4,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{32 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}-\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \sin (c+d x)}{7 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{32 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}-\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \sin (c+d x)}{7 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"ArcTanh[Sin[c + d*x]]/(a^4*d) - (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (32*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) - Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3)","A",6,4,19,0.2105,1,"{2766, 2978, 12, 3770}"
80,1,135,0,0.3912062,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^4,x]","\frac{664 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{88 \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{12 \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}","\frac{664 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{88 \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{12 \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-4*ArcTanh[Sin[c + d*x]])/(a^4*d) + (664*Tan[c + d*x])/(105*a^4*d) - (88*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - Tan[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (12*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,21,0.2857,1,"{2766, 2978, 2748, 3767, 8, 3770}"
81,1,185,0,0.431702,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^4,x]","-\frac{576 \tan (c+d x)}{35 a^4 d}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{288 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{43 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}","-\frac{576 \tan (c+d x)}{35 a^4 d}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{288 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{43 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(21*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (576*Tan[c + d*x])/(35*a^4*d) + (21*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - (43*Sec[c + d*x]*Tan[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) - (288*Sec[c + d*x]*Tan[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",9,7,21,0.3333,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
82,1,225,0,0.5155352,"\int \frac{\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5,x]","-\frac{7664 \sin (c+d x)}{315 a^5 d}-\frac{28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac{31 x}{2 a^5}-\frac{\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4}","-\frac{7664 \sin (c+d x)}{315 a^5 d}-\frac{28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac{31 x}{2 a^5}-\frac{\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"(31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",6,3,21,0.1429,1,"{2765, 2977, 2734}"
83,1,191,0,0.4902091,"\int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5,x]","\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}+\frac{5 \sin (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{5 x}{a^5}-\frac{\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4}","\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}+\frac{5 \sin (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{5 x}{a^5}-\frac{\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4}",1,"(-5*x)/a^5 + (181*Sin[c + d*x])/(63*a^5*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (5*Cos[c + d*x]^4*Sin[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Cos[c + d*x]^2*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))","A",9,7,21,0.3333,1,"{2765, 2977, 2968, 3023, 12, 2735, 2648}"
84,1,168,0,0.3921322,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5,x]","-\frac{34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{661 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{x}{a^5}-\frac{\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4}","-\frac{34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{661 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{x}{a^5}-\frac{\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (173*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (661*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",7,6,21,0.2857,1,"{2765, 2977, 2968, 3019, 2735, 2648}"
85,1,155,0,0.3000666,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5,x]","\frac{83 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4}","\frac{83 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"-(Cos[c + d*x]^3*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (11*Cos[c + d*x]^2*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (67*Sin[c + d*x])/(315*a^2*d*(a + a*Cos[c + d*x])^3) - (142*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) + (83*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2765, 2977, 2968, 3019, 2750, 2648}"
86,1,147,0,0.2284387,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]","\frac{5 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{5 \sin (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{17 \sin (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}+\frac{\sin (c+d x)}{7 a d (a \cos (c+d x)+a)^4}","\frac{5 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{5 \sin (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{17 \sin (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}+\frac{\sin (c+d x)}{7 a d (a \cos (c+d x)+a)^4}",1,"-(Cos[c + d*x]^2*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) + Sin[c + d*x]/(7*a*d*(a + a*Cos[c + d*x])^4) - (17*Sin[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) + (5*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(63*d*(a^5 + a^5*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2765, 2968, 3019, 2750, 2650, 2648}"
87,1,139,0,0.1444405,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]","\frac{2 \sin (c+d x)}{45 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{45 a^3 d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{15 a^2 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x)}{9 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}","\frac{2 \sin (c+d x)}{45 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{45 a^3 d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{15 a^2 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x)}{9 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (2*Sin[c + d*x])/(9*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(15*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(45*a^3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(45*d*(a^5 + a^5*Cos[c + d*x]))","A",5,4,21,0.1905,1,"{2758, 2750, 2650, 2648}"
88,1,143,0,0.1054706,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^5,x]","\frac{2 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{63 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{\sin (c+d x)}{21 a^2 d (a \cos (c+d x)+a)^3}+\frac{5 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}","\frac{2 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{63 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{\sin (c+d x)}{21 a^2 d (a \cos (c+d x)+a)^3}+\frac{5 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"-Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(21*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(63*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(63*d*(a^5 + a^5*Cos[c + d*x]))","A",5,3,19,0.1579,1,"{2750, 2650, 2648}"
89,1,143,0,0.0913571,"\int \frac{1}{(a+a \cos (c+d x))^5} \, dx","Int[(a + a*Cos[c + d*x])^(-5),x]","\frac{8 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{8 \sin (c+d x)}{315 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac{4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}","\frac{8 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{8 \sin (c+d x)}{315 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac{4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(315*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",5,2,12,0.1667,1,"{2650, 2648}"
90,1,153,0,0.3774813,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^5,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{488 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{34 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{13 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{488 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{34 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{13 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"ArcTanh[Sin[c + d*x]]/(a^5*d) - Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (13*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) - (173*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (488*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",7,4,19,0.2105,1,"{2766, 2978, 12, 3770}"
91,1,168,0,0.5328237,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]","\frac{496 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{5 \tan (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{67 \tan (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \tan (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{5 \tan (c+d x)}{21 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \cos (c+d x)+a)^5}","\frac{496 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{5 \tan (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{67 \tan (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \tan (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{5 \tan (c+d x)}{21 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(-5*ArcTanh[Sin[c + d*x]])/(a^5*d) + (496*Tan[c + d*x])/(63*a^5*d) - Tan[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (5*Tan[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Tan[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Tan[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) - (5*Tan[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))","A",9,6,21,0.2857,1,"{2766, 2978, 2748, 3767, 8, 3770}"
92,1,224,0,0.5396829,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]","-\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac{3832 \tan (c+d x) \sec (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}","-\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac{3832 \tan (c+d x) \sec (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(31*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (7664*Tan[c + d*x])/(315*a^5*d) + (31*Sec[c + d*x]*Tan[c + d*x])/(2*a^5*d) - (Sec[c + d*x]*Tan[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Sec[c + d*x]*Tan[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Sec[c + d*x]*Tan[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Sec[c + d*x]*Tan[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Sec[c + d*x]*Tan[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))","A",10,7,21,0.3333,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
93,1,184,0,0.4096259,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx","Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6,x]","-\frac{118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}+\frac{146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac{268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac{130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5}","-\frac{118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}+\frac{146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac{268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac{130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5}",1,"(130*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^3) - (268*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^2) + (146*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6) - (14*Cos[c + d*x]^3*Sin[c + d*x])/(99*a*d*(a + a*Cos[c + d*x])^5) - (118*Cos[c + d*x]^2*Sin[c + d*x])/(693*a^2*d*(a + a*Cos[c + d*x])^4)","A",7,6,21,0.2857,1,"{2765, 2977, 2968, 3019, 2750, 2648}"
94,1,176,0,0.3183566,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6,x]","\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac{241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac{9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5}","\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac{241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac{9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5}",1,"(-241*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^3) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^2) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(33*a*d*(a + a*Cos[c + d*x])^5) + (9*Sin[c + d*x])/(77*a^2*d*(a + a*Cos[c + d*x])^4)","A",7,7,21,0.3333,1,"{2765, 2977, 2968, 3019, 2750, 2650, 2648}"
95,1,158,0,0.2411813,"\int \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 a \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{32 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{64 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{32 a \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{32 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{64 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{32 a \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}",1,"(32*a*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (64*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (32*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)","A",5,4,23,0.1739,1,"{2770, 2759, 2751, 2646}"
96,1,122,0,0.1755563,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{4 a \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{4 a \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (8*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (12*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)","A",4,4,23,0.1739,1,"{2770, 2759, 2751, 2646}"
97,1,86,0,0.1146552,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{14 a \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}","\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{14 a \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(14*a*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",3,3,23,0.1304,1,"{2759, 2751, 2646}"
98,1,56,0,0.0456321,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",2,2,21,0.09524,1,"{2751, 2646}"
99,1,26,0,0.0129286,"\int \sqrt{a+a \cos (c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",1,1,14,0.07143,1,"{2646}"
100,1,37,0,0.0513787,"\int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d","A",2,2,21,0.09524,1,"{2773, 206}"
101,1,62,0,0.1036507,"\int \sqrt{a+a \cos (c+d x)} \sec ^2(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{a \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2772, 2773, 206}"
102,1,102,0,0.162247,"\int \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{3 a \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{3 a \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (3*a*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",4,3,23,0.1304,1,"{2772, 2773, 206}"
103,1,138,0,0.218053,"\int \sqrt{a+a \cos (c+d x)} \sec ^4(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^4,x]","\frac{5 a \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{5 a \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(5*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (5*a*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",5,3,23,0.1304,1,"{2772, 2773, 206}"
104,1,162,0,0.2474461,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{34 a^2 \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{68 a^2 \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{68 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}","\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{34 a^2 \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{68 a^2 \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{68 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}",1,"(68*a^2*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (34*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (136*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (68*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)","A",6,6,23,0.2609,1,"{2763, 21, 2770, 2759, 2751, 2646}"
105,1,116,0,0.1412817,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2),x]","\frac{152 a^2 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}","\frac{152 a^2 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}",1,"(152*a^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2647, 2646}"
106,1,86,0,0.0653159,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2),x]","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(8*a^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",3,3,21,0.1429,1,"{2751, 2647, 2646}"
107,1,59,0,0.0293274,"\int (a+a \cos (c+d x))^{3/2} \, dx","Int[(a + a*Cos[c + d*x])^(3/2),x]","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",2,2,14,0.1429,1,"{2647, 2646}"
108,1,66,0,0.1069579,"\int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x],x]","\frac{2 a^2 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 a^2 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,21,0.1905,1,"{2763, 21, 2773, 206}"
109,1,65,0,0.1180481,"\int (a+a \cos (c+d x))^{3/2} \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{a^2 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(3*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^2*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,23,0.1739,1,"{2762, 21, 2773, 206}"
110,1,106,0,0.175759,"\int (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3,x]","\frac{7 a^2 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","\frac{7 a^2 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(7*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (7*a^2*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2762, 21, 2772, 2773, 206}"
111,1,144,0,0.2366089,"\int (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4,x]","\frac{11 a^2 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{11 a^2 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(11*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (11*a^2*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,23,0.2174,1,"{2762, 21, 2772, 2773, 206}"
112,1,203,0,0.3629809,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2),x]","\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{46 a^3 \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{710 a^3 \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}-\frac{568 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{284 a^3 \sin (c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{284 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}","\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{46 a^3 \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{710 a^3 \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}-\frac{568 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{284 a^3 \sin (c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{284 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}",1,"(284*a^3*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (710*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (568*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a^2*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (284*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d)","A",6,6,23,0.2609,1,"{2763, 2981, 2770, 2759, 2751, 2646}"
113,1,146,0,0.1600445,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2),x]","\frac{208 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}","\frac{208 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}",1,"(832*a^3*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (26*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)","A",5,4,23,0.1739,1,"{2759, 2751, 2647, 2646}"
114,1,116,0,0.0869676,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2),x]","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(64*a^3*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",4,3,21,0.1429,1,"{2751, 2647, 2646}"
115,1,89,0,0.0500422,"\int (a+a \cos (c+d x))^{5/2} \, dx","Int[(a + a*Cos[c + d*x])^(5/2),x]","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",3,2,14,0.1429,1,"{2647, 2646}"
116,1,98,0,0.2022068,"\int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]","\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2763, 2981, 2773, 206}"
117,1,92,0,0.1981697,"\int (a+a \cos (c+d x))^{5/2} \sec ^2(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2,x]","\frac{a^3 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}+\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{a^3 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}+\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(5*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d","A",4,4,23,0.1739,1,"{2762, 2981, 2773, 206}"
118,1,106,0,0.2224097,"\int (a+a \cos (c+d x))^{5/2} \sec ^3(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3,x]","\frac{9 a^3 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{9 a^3 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(19*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (9*a^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,23,0.1739,1,"{2762, 2980, 2773, 206}"
119,1,144,0,0.2822934,"\int (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4,x]","\frac{25 a^3 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{25 a^3 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(25*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (25*a^3*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,23,0.2174,1,"{2762, 2980, 2772, 2773, 206}"
120,1,182,0,0.3445939,"\int (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5,x]","\frac{163 a^3 \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{163 a^3 \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}","\frac{163 a^3 \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{163 a^3 \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}",1,"(163*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (163*a^3*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,5,23,0.2174,1,"{2762, 2980, 2772, 2773, 206}"
121,1,119,0,0.0682629,"\int (a+a \cos (c+d x))^{7/2} \, dx","Int[(a + a*Cos[c + d*x])^(7/2),x]","\frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}","\frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",4,2,14,0.1429,1,"{2647, 2646}"
122,1,174,0,0.3715511,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{62 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{148 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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 \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{62 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{148 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (148*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (62*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)","A",7,7,23,0.3043,1,"{2778, 2983, 2968, 3023, 2751, 2649, 206}"
123,1,140,0,0.2392517,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{28 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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 \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{28 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (28*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)","A",6,6,23,0.2609,1,"{2778, 2968, 3023, 2751, 2649, 206}"
124,1,104,0,0.1252092,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2649, 206}"
125,1,73,0,0.0510623,"\int \frac{\cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,21,0.1429,1,"{2751, 2649, 206}"
126,1,46,0,0.0220375,"\int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[1/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)","A",2,2,14,0.1429,1,"{2649, 206}"
127,1,85,0,0.1142288,"\int \frac{\sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \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 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \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*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)","A",5,4,21,0.1905,1,"{2780, 2649, 206, 2773}"
128,1,108,0,0.2127886,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-(ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + Tan[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,23,0.2174,1,"{2779, 2985, 2649, 206, 2773}"
129,1,147,0,0.3402601,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \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} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}","-\frac{\tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \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} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - Tan[c + d*x]/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,23,0.2609,1,"{2779, 2984, 2985, 2649, 206, 2773}"
130,1,181,0,0.4886819,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]],x]","\frac{7 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}","\frac{7 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(-9*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (7*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,23,0.2609,1,"{2779, 2984, 2985, 2649, 206, 2773}"
131,1,183,0,0.3998338,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{15 \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{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}","-\frac{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{15 \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{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}",1,"(-15*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (31*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + (9*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - (13*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)","A",7,7,23,0.3043,1,"{2765, 2983, 2968, 3023, 2751, 2649, 206}"
132,1,145,0,0.2610796,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2),x]","\frac{7 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{11 \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{\sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}","\frac{7 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{11 \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{\sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"(11*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - (13*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + (7*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)","A",6,6,23,0.2609,1,"{2765, 2968, 3023, 2751, 2649, 206}"
133,1,105,0,0.1338435,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{7 \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 \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}+\frac{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{7 \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 \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}+\frac{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(-7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,23,0.1739,1,"{2758, 2751, 2649, 206}"
134,1,77,0,0.0590195,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{3 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{3 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",3,3,21,0.1429,1,"{2750, 2649, 206}"
135,1,77,0,0.0389913,"\int \frac{1}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Cos[c + d*x])^(-3/2),x]","\frac{\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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{\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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",3,3,14,0.2143,1,"{2650, 2649, 206}"
136,1,114,0,0.2208846,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,21,0.2381,1,"{2766, 2985, 2649, 206, 2773}"
137,1,144,0,0.3740431,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \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 \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \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 \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(-3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + (9*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,23,0.2609,1,"{2766, 2984, 2985, 2649, 206, 2773}"
138,1,185,0,0.4994309,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2),x]","\frac{19 \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 \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 \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{\tan (c+d x) \sec (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{19 \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 \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 \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{\tan (c+d x) \sec (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(19*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - (13*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (7*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (Sec[c + d*x]*Tan[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])","A",8,6,23,0.2609,1,"{2766, 2984, 2985, 2649, 206, 2773}"
139,1,183,0,0.4115396,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(5/2),x]","\frac{95 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{197 \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \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{\sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{17 \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{95 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{197 \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \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{\sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{17 \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(163*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - (197*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (95*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)","A",7,7,23,0.3043,1,"{2765, 2977, 2968, 3023, 2751, 2649, 206}"
140,1,145,0,0.2701858,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(5/2),x]","\frac{9 \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{75 \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{\sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{9 \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{75 \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{\sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-75*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (13*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (9*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,23,0.2609,1,"{2765, 2968, 3019, 2751, 2649, 206}"
141,1,107,0,0.1369255,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2),x]","\frac{19 \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{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{19 \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{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(19*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{2758, 2750, 2649, 206}"
142,1,107,0,0.080487,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{5 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{5 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,21,0.1905,1,"{2750, 2650, 2649, 206}"
143,1,107,0,0.0633022,"\int \frac{1}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Cos[c + d*x])^(-5/2),x]","\frac{3 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{3 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",4,3,14,0.2143,1,"{2650, 2649, 206}"
144,1,144,0,0.3347475,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \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{11 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \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{11 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,21,0.2857,1,"{2766, 2978, 2985, 2649, 206, 2773}"
145,1,174,0,0.5192349,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2),x]","\frac{35 \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \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 \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{35 \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \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 \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + (115*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,23,0.3043,1,"{2766, 2978, 2984, 2985, 2649, 206, 2773}"
146,1,111,0,0.0776239,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]),x]","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,4,21,0.1905,1,"{2748, 2635, 2639, 2641}"
147,1,87,0,0.0666751,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]),x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,4,21,0.1905,1,"{2748, 2635, 2641, 2639}"
148,1,61,0,0.0505184,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]),x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2748, 2639, 2635, 2641}"
149,1,35,0,0.0388943,"\int \frac{a+a \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/d","A",3,3,21,0.1429,1,"{2748, 2641, 2639}"
150,1,57,0,0.0496062,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])/Cos[c + d*x]^(3/2),x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/d + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,21,0.1905,1,"{2748, 2636, 2639, 2641}"
151,1,83,0,0.0594036,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])/Cos[c + d*x]^(5/2),x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,4,21,0.1905,1,"{2748, 2636, 2641, 2639}"
152,1,111,0,0.0739601,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])/Cos[c + d*x]^(7/2),x]","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,4,21,0.1905,1,"{2748, 2636, 2639, 2641}"
153,1,147,0,0.1379324,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2,x]","\frac{20 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{32 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{32 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{20 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{32 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{32 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(32*a^2*EllipticE[(c + d*x)/2, 2])/(15*d) + (20*a^2*EllipticF[(c + d*x)/2, 2])/(21*d) + (20*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",10,4,23,0.1739,1,"{2757, 2635, 2639, 2641}"
154,1,121,0,0.1186186,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2,x]","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}",1,"(12*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*EllipticF[(c + d*x)/2, 2])/(7*d) + (8*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (4*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",9,4,23,0.1739,1,"{2757, 2635, 2641, 2639}"
155,1,95,0,0.0932659,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2,x]","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(16*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",7,4,23,0.1739,1,"{2757, 2639, 2635, 2641}"
156,1,67,0,0.0807783,"\int \frac{(a+a \cos (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(4*a^2*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,4,23,0.1739,1,"{2757, 2641, 2639, 2635}"
157,1,44,0,0.0804822,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(3/2),x]","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(4*a^2*EllipticF[(c + d*x)/2, 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,4,23,0.1739,1,"{2757, 2636, 2639, 2641}"
158,1,91,0,0.092857,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-4*a^2*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,4,23,0.1739,1,"{2757, 2636, 2641, 2639}"
159,1,121,0,0.1185145,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(7/2),x]","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-16*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (16*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",9,4,23,0.1739,1,"{2757, 2636, 2639, 2641}"
160,1,147,0,0.1528084,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3,x]","\frac{44 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{68 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{68 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{44 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{44 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{68 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{68 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{44 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(68*a^3*EllipticE[(c + d*x)/2, 2])/(15*d) + (44*a^3*EllipticF[(c + d*x)/2, 2])/(21*d) + (44*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (68*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (6*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^3*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",12,4,23,0.1739,1,"{2757, 2635, 2641, 2639}"
161,1,121,0,0.1284613,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3,x]","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(28*a^3*EllipticE[(c + d*x)/2, 2])/(5*d) + (52*a^3*EllipticF[(c + d*x)/2, 2])/(21*d) + (52*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (6*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",10,4,23,0.1739,1,"{2757, 2639, 2635, 2641}"
162,1,91,0,0.1090217,"\int \frac{(a+a \cos (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}",1,"(36*a^3*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*EllipticF[(c + d*x)/2, 2])/d + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",8,4,23,0.1739,1,"{2757, 2641, 2639, 2635}"
163,1,91,0,0.1103646,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(4*a^3*EllipticE[(c + d*x)/2, 2])/d + (20*a^3*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,5,23,0.2174,1,"{2757, 2636, 2639, 2641, 2635}"
164,1,91,0,0.1048927,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(5/2),x]","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-4*a^3*EllipticE[(c + d*x)/2, 2])/d + (20*a^3*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,4,23,0.1739,1,"{2757, 2636, 2641, 2639}"
165,1,117,0,0.1271164,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(7/2),x]","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{36 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{36 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-36*a^3*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*EllipticF[(c + d*x)/2, 2])/d + (2*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a^3*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (36*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",10,4,23,0.1739,1,"{2757, 2636, 2639, 2641}"
166,1,147,0,0.1481104,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(9/2),x]","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{52 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{28 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{52 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{28 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-28*a^3*EllipticE[(c + d*x)/2, 2])/(5*d) + (52*a^3*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (52*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (28*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",12,4,23,0.1739,1,"{2757, 2636, 2641, 2639}"
167,1,173,0,0.2045901,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4,x]","\frac{904 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{150 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{128 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{904 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}","\frac{904 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{150 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{128 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{904 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}",1,"(128*a^4*EllipticE[(c + d*x)/2, 2])/(15*d) + (904*a^4*EllipticF[(c + d*x)/2, 2])/(231*d) + (904*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (128*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (150*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (8*a^4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a^4*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)","A",16,4,23,0.1739,1,"{2757, 2635, 2641, 2639}"
168,1,147,0,0.1637815,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4,x]","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}",1,"(152*a^4*EllipticE[(c + d*x)/2, 2])/(15*d) + (32*a^4*EllipticF[(c + d*x)/2, 2])/(7*d) + (32*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (122*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (8*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",13,4,23,0.1739,1,"{2757, 2639, 2635, 2641}"
169,1,121,0,0.1397955,"\int \frac{(a+a \cos (c+d x))^4}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^4/Sqrt[Cos[c + d*x]],x]","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(64*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (136*a^4*EllipticF[(c + d*x)/2, 2])/(21*d) + (94*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (8*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",11,4,23,0.1739,1,"{2757, 2641, 2639, 2635}"
170,1,119,0,0.1221097,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(3/2),x]","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(56*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (32*a^4*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (8*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",10,5,23,0.2174,1,"{2757, 2636, 2639, 2641, 2635}"
171,1,98,0,0.1224759,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(5/2),x]","\frac{40 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{40 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(40*a^4*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*a^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",10,5,23,0.2174,1,"{2757, 2636, 2641, 2639, 2635}"
172,1,121,0,0.1464172,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(7/2),x]","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{66 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{66 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-56*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (32*a^4*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*a^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (66*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",11,4,23,0.1739,1,"{2757, 2636, 2639, 2641}"
173,1,147,0,0.1665292,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(9/2),x]","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-64*a^4*EllipticE[(c + d*x)/2, 2])/(5*d) + (136*a^4*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (94*a^4*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (64*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",13,4,23,0.1739,1,"{2757, 2636, 2641, 2639}"
174,1,128,0,0.1096113,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x]),x]","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(21*EllipticE[(c + d*x)/2, 2])/(5*a*d) - (5*EllipticF[(c + d*x)/2, 2])/(3*a*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2767, 2748, 2635, 2641, 2639}"
175,1,100,0,0.1014559,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x]),x]","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(-3*EllipticE[(c + d*x)/2, 2])/(a*d) + (5*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,23,0.2174,1,"{2767, 2748, 2639, 2635, 2641}"
176,1,72,0,0.0863407,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x]),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(3*EllipticE[(c + d*x)/2, 2])/(a*d) - EllipticF[(c + d*x)/2, 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,23,0.1739,1,"{2767, 2748, 2641, 2639}"
177,1,70,0,0.0825385,"\int \frac{\sqrt{\cos (c+d x)}}{a+a \cos (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x]),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"-(EllipticE[(c + d*x)/2, 2]/(a*d)) + EllipticF[(c + d*x)/2, 2]/(a*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,23,0.1739,1,"{2769, 2748, 2641, 2639}"
178,1,70,0,0.0851393,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"EllipticE[(c + d*x)/2, 2]/(a*d) + EllipticF[(c + d*x)/2, 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",4,4,23,0.1739,1,"{2768, 2748, 2641, 2639}"
179,1,96,0,0.0987303,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"(-3*EllipticE[(c + d*x)/2, 2])/(a*d) - EllipticF[(c + d*x)/2, 2]/(a*d) + (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",5,5,23,0.2174,1,"{2768, 2748, 2636, 2639, 2641}"
180,1,124,0,0.1120244,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(3*EllipticE[(c + d*x)/2, 2])/(a*d) + (5*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2768, 2748, 2636, 2641, 2639}"
181,1,160,0,0.2191121,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(56*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*EllipticF[(c + d*x)/2, 2])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d) + (56*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - (3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,6,23,0.2609,1,"{2765, 2977, 2748, 2635, 2641, 2639}"
182,1,138,0,0.201635,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^2,x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-7*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (10*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (10*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,23,0.2609,1,"{2765, 2977, 2748, 2639, 2635, 2641}"
183,1,112,0,0.1836981,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(4*EllipticE[(c + d*x)/2, 2])/(a^2*d) - (5*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,23,0.2174,1,"{2765, 2977, 2748, 2641, 2639}"
184,1,109,0,0.1878337,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2,x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"-(EllipticE[(c + d*x)/2, 2]/(a^2*d)) + (2*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,23,0.2174,1,"{2765, 2978, 2748, 2641, 2639}"
185,1,57,0,0.0544056,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^2,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"EllipticF[(c + d*x)/2, 2]/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",3,3,23,0.1304,1,"{2764, 21, 2641}"
186,1,109,0,0.1838579,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"EllipticE[(c + d*x)/2, 2]/(a^2*d) + (2*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",5,5,23,0.2174,1,"{2766, 2978, 2748, 2641, 2639}"
187,1,136,0,0.2079496,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"(-4*EllipticE[(c + d*x)/2, 2])/(a^2*d) - (5*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (4*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (5*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",6,6,23,0.2609,1,"{2766, 2978, 2748, 2636, 2639, 2641}"
188,1,162,0,0.2390735,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{10 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{10 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(7*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (10*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - (7*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",7,6,23,0.2609,1,"{2766, 2978, 2748, 2636, 2641, 2639}"
189,1,207,0,0.3351909,"\int \frac{\cos ^{\frac{11}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(11/2)/(a + a*Cos[c + d*x])^3,x]","-\frac{21 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{231 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2}","-\frac{21 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{231 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2}",1,"(231*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (21*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (63*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",8,6,23,0.2609,1,"{2765, 2977, 2748, 2635, 2641, 2639}"
190,1,181,0,0.322797,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^3,x]","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(-119*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + (11*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (119*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,23,0.2609,1,"{2765, 2977, 2748, 2639, 2635, 2641}"
191,1,155,0,0.2986011,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^3,x]","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{8 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{8 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(49*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (13*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (8*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2765, 2977, 2748, 2641, 2639}"
192,1,155,0,0.3077417,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}",1,"(-9*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(2*a^3*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,6,23,0.2609,1,"{2765, 2977, 2978, 2748, 2641, 2639}"
193,1,155,0,0.303611,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"-EllipticE[(c + d*x)/2, 2]/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(6*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2765, 2978, 2748, 2641, 2639}"
194,1,155,0,0.3052656,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^3,x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"EllipticE[(c + d*x)/2, 2]/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(6*a^3*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2764, 2978, 2748, 2641, 2639}"
195,1,155,0,0.3119089,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(9*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,23,0.2174,1,"{2766, 2978, 2748, 2641, 2639}"
196,1,181,0,0.3472493,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"(-49*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (13*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + (49*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",7,6,23,0.2609,1,"{2766, 2978, 2748, 2636, 2639, 2641}"
197,1,207,0,0.3590687,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(119*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + (11*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",8,6,23,0.2609,1,"{2766, 2978, 2748, 2636, 2641, 2639}"
198,1,154,0,0.2330183,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]],x]","\frac{a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{5 a \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}","\frac{a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{5 a \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}",1,"(5*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (5*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",5,3,25,0.1200,1,"{2770, 2774, 216}"
199,1,116,0,0.1743063,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]],x]","\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}","\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(3*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",4,3,25,0.1200,1,"{2770, 2774, 216}"
200,1,72,0,0.116369,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2770, 2774, 216}"
201,1,37,0,0.058023,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d","A",2,2,25,0.08000,1,"{2774, 216}"
202,1,36,0,0.0567769,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",1,1,25,0.04000,1,"{2771}"
203,1,77,0,0.1089444,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2772, 2771}"
204,1,115,0,0.1652832,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",3,2,25,0.08000,1,"{2772, 2771}"
205,1,153,0,0.2275987,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]","\frac{16 a \sin (c+d x)}{35 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{16 a \sin (c+d x)}{35 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (32*a*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,2,25,0.08000,1,"{2772, 2771}"
206,1,160,0,0.2480811,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{11 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}","\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{11 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}",1,"(11*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (11*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,25,0.2000,1,"{2763, 21, 2770, 2774, 216}"
207,1,120,0,0.1849098,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{7 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}","\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{7 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(7*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (7*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2763, 21, 2770, 2774, 216}"
208,1,75,0,0.1197023,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\frac{3 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{3 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(3*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2763, 21, 2774, 216}"
209,1,76,0,0.1242069,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{2 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{2 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2762, 21, 2774, 216}"
210,1,81,0,0.1180349,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (10*a^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2762, 21, 2771}"
211,1,121,0,0.1732414,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]","\frac{6 a^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{6 a^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (6*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (12*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2762, 21, 2772, 2771}"
212,1,161,0,0.2346184,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]","\frac{104 a^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{104 a^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,4,25,0.1600,1,"{2762, 21, 2772, 2771}"
213,1,200,0,0.3581233,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2),x]","\frac{17 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{163 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{163 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}","\frac{17 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{163 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{163 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}",1,"(163*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (163*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,25,0.2000,1,"{2763, 2981, 2770, 2774, 216}"
214,1,160,0,0.2952909,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2),x]","\frac{13 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{25 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{25 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}","\frac{13 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{25 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{25 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}",1,"(25*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (25*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{2763, 2981, 2770, 2774, 216}"
215,1,120,0,0.2315615,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{19 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}","\frac{19 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(19*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (9*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,25,0.1600,1,"{2763, 2981, 2774, 216}"
216,1,114,0,0.2249576,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{5 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}","\frac{5 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(5*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2762, 2981, 2774, 216}"
217,1,118,0,0.225688,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,25,0.1600,1,"{2762, 2980, 2774, 216}"
218,1,121,0,0.2245875,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(7/2),x]","\frac{22 a^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{86 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{22 a^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{86 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(22*a^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (86*a^3*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",3,3,25,0.1200,1,"{2762, 2980, 2771}"
219,1,161,0,0.2860091,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(9/2),x]","\frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(6*a^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (92*a^3*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",4,4,25,0.1600,1,"{2762, 2980, 2772, 2771}"
220,1,201,0,0.3506753,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(11/2),x]","\frac{584 a^3 \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{584 a^3 \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(38*a^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (1168*a^3*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",5,4,25,0.1600,1,"{2762, 2980, 2772, 2771}"
221,1,38,0,0.0561667,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{4}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/4),x]","\frac{4 a^2 \sin (c+d x)}{d \sqrt[4]{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{4 a^2 \sin (c+d x)}{d \sqrt[4]{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(4*a^2*Sin[c + d*x])/(d*Cos[c + d*x]^(1/4)*Sqrt[a + a*Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2762, 8}"
222,1,37,0,0.0597373,"\int \frac{\sqrt{a+a \cos (e+f x)}}{\sqrt{\cos (e+f x)}} \, dx","Int[Sqrt[a + a*Cos[e + f*x]]/Sqrt[Cos[e + f*x]],x]","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a \cos (e+f x)+a}}\right)}{f}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a \cos (e+f x)+a}}\right)}{f}",1,"(2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + a*Cos[e + f*x]]])/f","A",2,2,25,0.08000,1,"{2774, 216}"
223,1,38,0,0.0721399,"\int \frac{\sqrt{a-a \cos (e+f x)}}{\sqrt{-\cos (e+f x)}} \, dx","Int[Sqrt[a - a*Cos[e + f*x]]/Sqrt[-Cos[e + f*x]],x]","-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a-a \cos (e+f x)}}\right)}{f}","-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a-a \cos (e+f x)}}\right)}{f}",1,"(-2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a - a*Cos[e + f*x]]])/f","A",2,2,28,0.07143,1,"{2774, 216}"
224,1,171,0,0.4203484,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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,"(7*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2778, 2983, 2982, 2782, 205, 2774, 216}"
225,1,128,0,0.2799743,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tan ^{-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{\sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tan ^{-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,"-(ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2778, 2982, 2782, 205, 2774, 216}"
226,1,95,0,0.1698148,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-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 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-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,"(2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*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",5,5,25,0.2000,1,"{2777, 2774, 216, 2782, 205}"
227,1,56,0,0.0617989,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\sqrt{2} \tan ^{-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{\sqrt{2} \tan ^{-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]*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",2,2,25,0.08000,1,"{2782, 205}"
228,1,93,0,0.1278287,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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]*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*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2779, 12, 2782, 205}"
229,1,131,0,0.2357232,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tan ^{-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 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tan ^{-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]*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*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2779, 2984, 12, 2782, 205}"
230,1,169,0,0.3648641,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-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]*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*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,25,0.2000,1,"{2779, 2984, 12, 2782, 205}"
231,1,126,0,0.2722625,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{7 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{\cos (c+d x)+1}}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{7 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{\cos (c+d x)+1}}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (7*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]])/(4*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 + Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Cos[c + d*x]])","A",7,6,23,0.2609,1,"{2778, 2983, 2982, 2781, 216, 2774}"
232,1,85,0,0.1873215,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{\cos (c+d x)+1}}","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{\cos (c+d x)+1}}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d - ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])","A",6,5,23,0.2174,1,"{2778, 2982, 2781, 216, 2774}"
233,1,54,0,0.1166531,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[1 + Cos[c + d*x]],x]","\frac{2 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{2 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]])/d","A",5,4,23,0.1739,1,"{2777, 2774, 216, 2781}"
234,1,27,0,0.0428342,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]),x]","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d","A",2,2,23,0.08696,1,"{2781, 216}"
235,1,62,0,0.0845289,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2779, 2781, 216}"
236,1,98,0,0.1651761,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[1 + Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2779, 2984, 12, 2781, 216}"
237,1,134,0,0.2424718,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*Sqrt[1 + Cos[c + d*x]]),x]","-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}","-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])","A",6,5,23,0.2174,1,"{2779, 2984, 12, 2781, 216}"
238,1,174,0,0.4245412,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{3 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}","-\frac{3 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"(-3*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + (9*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) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2765, 2983, 2982, 2782, 205, 2774, 216}"
239,1,134,0,0.2922401,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - (5*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,6,25,0.2400,1,"{2765, 2982, 2782, 205, 2774, 216}"
240,1,97,0,0.1303382,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{\tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"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) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{2764, 12, 2782, 205}"
241,1,97,0,0.1294799,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{3 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","\frac{3 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(3*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{2766, 12, 2782, 205}"
242,1,137,0,0.2469467,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{7 \tan ^{-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 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}","-\frac{7 \tan ^{-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 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(-7*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) - Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (5*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2766, 2984, 12, 2782, 205}"
243,1,177,0,0.3881226,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{11 \tan ^{-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 \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{11 \tan ^{-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 \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(11*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) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (7*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (19*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,5,25,0.2000,1,"{2766, 2984, 12, 2782, 205}"
244,1,214,0,0.5765217,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{5 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{35 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{115 \tan ^{-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{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}","-\frac{5 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{35 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{115 \tan ^{-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{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-5*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + (115*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) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,8,25,0.3200,1,"{2765, 2977, 2983, 2982, 2782, 205, 2774, 216}"
245,1,174,0,0.4396436,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - (43*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) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",7,7,25,0.2800,1,"{2765, 2977, 2982, 2782, 205, 2774, 216}"
246,1,137,0,0.2588754,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{3 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{3 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(3*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{2765, 2978, 12, 2782, 205}"
247,1,137,0,0.2498346,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{5 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{5 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(5*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) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{2764, 2978, 12, 2782, 205}"
248,1,137,0,0.2605414,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{19 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{19 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(19*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{2766, 2978, 12, 2782, 205}"
249,1,177,0,0.4027338,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{49 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{75 \tan ^{-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 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{49 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{75 \tan ^{-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 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(-75*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) - Sin[c + d*x]/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2766, 2978, 2984, 12, 2782, 205}"
250,1,217,0,0.5461445,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{95 \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{163 \tan ^{-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 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","\frac{95 \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{163 \tan ^{-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 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(163*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) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (17*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (95*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (299*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,6,25,0.2400,1,"{2766, 2978, 2984, 12, 2782, 205}"
251,1,254,0,0.7485771,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(7/2),x]","-\frac{259 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{189 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{637 \tan ^{-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{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{7 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}","-\frac{259 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{189 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{637 \tan ^{-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{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{7 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}",1,"(-7*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) + (637*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) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (7*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (259*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (189*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",9,8,25,0.3200,1,"{2765, 2977, 2983, 2982, 2782, 205, 2774, 216}"
252,1,214,0,0.6011261,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{49 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{177 \tan ^{-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{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{17 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{49 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{177 \tan ^{-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{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{17 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"(2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) - (177*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) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (17*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (49*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",8,7,25,0.2800,1,"{2765, 2977, 2982, 2782, 205, 2774, 216}"
253,1,177,0,0.4036435,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{67 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{5 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}","\frac{67 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{5 \tan ^{-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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"(5*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) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + (67*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,6,25,0.2400,1,"{2765, 2977, 2978, 12, 2782, 205}"
254,1,177,0,0.4027145,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{17 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{7 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","\frac{17 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{7 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(7*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + (17*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,25,0.2000,1,"{2765, 2978, 12, 2782, 205}"
255,1,177,0,0.4031755,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(7/2),x]","-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{13 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{13 \tan ^{-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{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(13*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) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,25,0.2000,1,"{2764, 2978, 12, 2782, 205}"
256,1,177,0,0.4100817,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)),x]","-\frac{103 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{63 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","-\frac{103 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{63 \tan ^{-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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(63*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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (103*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))","A",6,5,25,0.2000,1,"{2766, 2978, 12, 2782, 205}"
257,1,217,0,0.5469122,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{691 \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{363 \tan ^{-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 \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}","\frac{691 \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{363 \tan ^{-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 \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(-363*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) - Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - (19*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (199*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (691*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",7,6,25,0.2400,1,"{2766, 2978, 2984, 12, 2782, 205}"
258,1,257,0,0.703447,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{193 \sin (c+d x)}{64 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{109 \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{629 \sin (c+d x)}{64 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{1015 \tan ^{-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 \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","\frac{193 \sin (c+d x)}{64 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{109 \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{629 \sin (c+d x)}{64 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{1015 \tan ^{-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 \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(1015*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) - Sin[c + d*x]/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - (23*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (109*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (193*Sin[c + d*x])/(64*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (629*Sin[c + d*x])/(64*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])","A",8,6,25,0.2400,1,"{2766, 2978, 2984, 12, 2782, 205}"
259,1,217,0,0.5569244,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx","Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]","\frac{853 \sin (c+d x) \sqrt{\cos (c+d x)}}{3072 a^3 d (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x) \sqrt{\cos (c+d x)}}{768 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{35 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{19 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 a d (a \cos (c+d x)+a)^{7/2}}","\frac{853 \sin (c+d x) \sqrt{\cos (c+d x)}}{3072 a^3 d (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x) \sqrt{\cos (c+d x)}}{768 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{35 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{19 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 a d (a \cos (c+d x)+a)^{7/2}}",1,"(35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (19*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)) - (187*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (853*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,25,0.2400,1,"{2765, 2977, 2978, 12, 2782, 205}"
260,1,217,0,0.5708406,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2),x]","\frac{73 \sin (c+d x) \sqrt{\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x) \sqrt{\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{45 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}}","\frac{73 \sin (c+d x) \sqrt{\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x) \sqrt{\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{45 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}}",1,"(45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)) + (33*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (73*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))","A",7,6,25,0.2400,1,"{2765, 2977, 2978, 12, 2782, 205}"
261,1,16,0,0.0433364,"\int \frac{1}{\sqrt{\cos (x)} \sqrt{1+\cos (x)}} \, dx","Int[1/(Sqrt[Cos[x]]*Sqrt[1 + Cos[x]]),x]","\sqrt{2} \sin ^{-1}\left(\frac{\sin (x)}{\cos (x)+1}\right)","\sqrt{2} \sin ^{-1}\left(\frac{\sin (x)}{\cos (x)+1}\right)",1,"Sqrt[2]*ArcSin[Sin[x]/(1 + Cos[x])]","A",2,2,15,0.1333,1,"{2781, 216}"
262,1,41,0,0.060865,"\int \frac{1}{\sqrt{\cos (x)} \sqrt{a+a \cos (x)}} \, dx","Int[1/(Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]]),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{\cos (x)} \sqrt{a \cos (x)+a}}\right)}{\sqrt{a}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{\cos (x)} \sqrt{a \cos (x)+a}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*ArcTan[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]])])/Sqrt[a]","A",2,2,17,0.1176,1,"{2782, 205}"
263,1,129,0,0.1917583,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]],x]","-\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 d}","-\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 d}",1,"(-3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*d) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) - (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])","A",4,3,26,0.1154,1,"{2770, 2775, 207}"
264,1,85,0,0.1217548,"\int \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]],x]","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}-\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}-\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}",1,"(Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d - (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])","A",3,3,26,0.1154,1,"{2770, 2775, 207}"
265,1,48,0,0.0661367,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[a - a*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d","A",2,2,26,0.07692,1,"{2775, 207}"
266,1,37,0,0.0566217,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",1,1,26,0.03846,1,"{2771}"
267,1,79,0,0.1158551,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) - (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",2,2,26,0.07692,1,"{2772, 2771}"
268,1,118,0,0.1726021,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","-\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}","-\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) - (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",3,2,26,0.07692,1,"{2772, 2771}"
269,1,114,0,0.1537228,"\int \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2),x]","-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}-\frac{3 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}","-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}-\frac{3 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}",1,"(-3*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])","A",4,3,25,0.1200,1,"{2770, 2775, 207}"
270,1,72,0,0.0859657,"\int \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)} \, dx","Int[Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}","\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}",1,"ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2770, 2775, 207}"
271,1,37,0,0.0433369,"\int \frac{\sqrt{1-\cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[1 - Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","-\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(-2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d","A",2,2,25,0.08000,1,"{2775, 207}"
272,1,35,0,0.0424697,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",1,1,25,0.04000,1,"{2771}"
273,1,75,0,0.0845189,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) - (4*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2772, 2771}"
274,1,112,0,0.131729,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","-\frac{8 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{5 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}","-\frac{8 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{5 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*Sin[c + d*x])/(5*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2)) - (8*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (16*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",3,2,25,0.08000,1,"{2772, 2771}"
275,1,185,0,0.4451109,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a-a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[a - a*Cos[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"(7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])","A",7,7,26,0.2692,1,"{2778, 2983, 2982, 2782, 208, 2775, 207}"
276,1,141,0,0.2954144,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a-a \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[a - a*Cos[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])]/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])","A",6,6,26,0.2308,1,"{2778, 2982, 2782, 208, 2775, 207}"
277,1,107,0,0.1804145,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a-a \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[a - a*Cos[c + d*x]],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"(2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)","A",5,5,26,0.1923,1,"{2777, 2775, 207, 2782, 208}"
278,1,58,0,0.0670286,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]]),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d))","A",2,2,26,0.07692,1,"{2782, 208}"
279,1,95,0,0.1327504,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",4,4,26,0.1538,1,"{2779, 12, 2782, 208}"
280,1,135,0,0.2505423,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",5,5,26,0.1923,1,"{2779, 2984, 12, 2782, 208}"
281,1,173,0,0.4008896,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","A",6,5,26,0.1923,1,"{2779, 2984, 12, 2782, 208}"
282,1,161,0,0.3016118,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[1 - Cos[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(7*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d) - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2778, 2983, 2982, 2782, 206, 2775, 207}"
283,1,118,0,0.213549,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[1 - Cos[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2778, 2982, 2782, 206, 2775, 207}"
284,1,85,0,0.1294914,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{1-\cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[1 - Cos[c + d*x]],x]","\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d","A",5,5,25,0.2000,1,"{2777, 2775, 207, 2782, 206}"
285,1,47,0,0.0482254,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Int[1/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"-((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d)","A",2,2,25,0.08000,1,"{2782, 206}"
286,1,83,0,0.0925313,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/(Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"-((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2779, 2782, 206}"
287,1,122,0,0.1823906,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/(Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2)),x]","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"-((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2779, 2984, 12, 2782, 206}"
288,1,78,0,0.1079798,"\int \cos ^{\frac{4}{3}}(c+d x) \sqrt[3]{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3),x]","\frac{2^{5/6} \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} F_1\left(\frac{1}{2};-\frac{4}{3},\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{5/6}}","\frac{2^{5/6} \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} F_1\left(\frac{1}{2};-\frac{4}{3},\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{5/6}}",1,"(2^(5/6)*AppellF1[1/2, -4/3, 1/6, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(5/6))","A",3,3,25,0.1200,1,"{2787, 2785, 133}"
289,1,79,0,0.1218348,"\int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","Int[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3),x]","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}",1,"(2*2^(1/6)*AppellF1[1/2, -4/3, -1/6, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))","A",3,3,25,0.1200,1,"{2787, 2785, 133}"
290,1,79,0,0.1208579,"\int \cos ^{\frac{5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","Int[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3),x]","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{5}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{5}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}",1,"(2*2^(1/6)*AppellF1[1/2, -5/3, -1/6, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))","A",3,3,25,0.1200,1,"{2787, 2785, 133}"
291,1,151,0,0.110528,"\int (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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 d}-\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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 d}-\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(-6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,6,21,0.2857,1,"{3238, 3787, 3768, 3771, 2641, 2639}"
292,1,123,0,0.097052,"\int (a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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 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)}{d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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 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)}{d}",1,"(-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,6,21,0.2857,1,"{3238, 3787, 3768, 3771, 2639, 2641}"
293,1,97,0,0.0872682,"\int (a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{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)}{d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{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)}{d}",1,"(-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,21,0.2857,1,"{3238, 3787, 3771, 2641, 3768, 2639}"
294,1,75,0,0.0757447,"\int (a+a \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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)}{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)}{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)}{d}",1,"(2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d","A",6,5,21,0.2381,1,"{3238, 3787, 3771, 2639, 2641}"
295,1,101,0,0.0899282,"\int \frac{a+a \cos (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])/Sqrt[Sec[c + d*x]],x]","\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 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)}{d}","\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 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)}{d}",1,"(2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2641, 2639}"
296,1,127,0,0.1003781,"\int \frac{a+a \cos (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])/Sec[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2639, 2641}"
297,1,151,0,0.1127729,"\int \frac{a+a \cos (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 a \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 \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)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 a \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 \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*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*a*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2641, 2639}"
298,1,161,0,0.146726,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{16 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 \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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{16 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 \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,"(-16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,7,23,0.3043,1,"{3238, 3788, 3768, 3771, 2641, 4046, 2639}"
299,1,131,0,0.1335782,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^2 \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 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^2 \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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,23,0.3043,1,"{3238, 3788, 3768, 3771, 2639, 4046, 2641}"
300,1,64,0,0.1076159,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 a^2 \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^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,23,0.2174,1,"{3238, 3788, 3771, 2641, 4043}"
301,1,107,0,0.1223858,"\int (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \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 \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 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,23,0.2609,1,"{3238, 3788, 3771, 2639, 4045, 2641}"
302,1,135,0,0.1379188,"\int \frac{(a+a \cos (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 \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 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 \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,"(16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,23,0.3043,1,"{3238, 3788, 3769, 3771, 2641, 4045, 2639}"
303,1,161,0,0.1505337,"\int \frac{(a+a \cos (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{4 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{4 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 \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,"(12*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])","A",9,7,23,0.3043,1,"{3238, 3788, 3769, 3771, 2639, 4045, 2641}"
304,1,187,0,0.2334057,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \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{28 a^3 \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^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \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{28 a^3 \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,"(-28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (28*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (52*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (6*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",17,6,23,0.2609,1,"{3238, 3791, 3768, 3771, 2639, 2641}"
305,1,157,0,0.2058864,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2),x]","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d}+\frac{36 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 \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^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d}+\frac{36 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 \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,"(-36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (36*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",15,6,23,0.2609,1,"{3238, 3791, 3771, 2641, 3768, 2639}"
306,1,131,0,0.1798102,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2),x]","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{20 a^3 \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 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{20 a^3 \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 \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^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",13,6,23,0.2609,1,"{3238, 3791, 3771, 2639, 2641, 3768}"
307,1,131,0,0.1764383,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2),x]","\frac{2 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{20 a^3 \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 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{20 a^3 \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 \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^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",13,7,23,0.3043,1,"{3238, 3791, 3769, 3771, 2641, 2639, 3768}"
308,1,131,0,0.1780928,"\int (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]],x]","\frac{2 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 \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^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 \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,"(36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])","A",13,6,23,0.2609,1,"{3238, 3791, 3769, 3771, 2639, 2641}"
309,1,161,0,0.2087606,"\int \frac{(a+a \cos (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{6 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{52 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{52 a^3 \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{28 a^3 \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{6 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{52 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{52 a^3 \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{28 a^3 \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,"(28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (52*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",15,6,23,0.2609,1,"{3238, 3791, 3769, 3771, 2641, 2639}"
310,1,187,0,0.2346379,"\int \frac{(a+a \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \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{68 a^3 \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{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \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{68 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(68*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (44*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (68*a^3*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (44*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",17,6,23,0.2609,1,"{3238, 3791, 3769, 3771, 2639, 2641}"
311,1,187,0,0.2532342,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(9/2),x]","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{64 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{136 a^4 \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{64 a^4 \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^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{64 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{136 a^4 \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{64 a^4 \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,"(-64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (64*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (94*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (8*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",19,6,23,0.2609,1,"{3238, 3791, 3771, 2641, 3768, 2639}"
312,1,161,0,0.2266442,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(7/2),x]","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{66 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{32 a^4 \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{56 a^4 \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^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{66 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{32 a^4 \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{56 a^4 \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,"(-56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (66*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",17,6,23,0.2609,1,"{3238, 3791, 3771, 2639, 2641, 3768}"
313,1,118,0,0.2106066,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(5/2),x]","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \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^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \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,"(40*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",16,7,23,0.3043,1,"{3238, 3791, 3769, 3771, 2641, 2639, 3768}"
314,1,159,0,0.2115466,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(3/2),x]","\frac{2 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \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{56 a^4 \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^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \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{56 a^4 \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,"(56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",16,7,23,0.3043,1,"{3238, 3791, 3769, 3771, 2639, 2641, 3768}"
315,1,161,0,0.227094,"\int (a+a \cos (c+d x))^4 \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]],x]","\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{136 a^4 \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{64 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{136 a^4 \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{64 a^4 \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,"(64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (8*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (94*a^4*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",17,6,23,0.2609,1,"{3238, 3791, 3769, 3771, 2641, 2639}"
316,1,187,0,0.2554775,"\int \frac{(a+a \cos (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^4/Sqrt[Sec[c + d*x]],x]","\frac{122 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{32 a^4 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 \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{122 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{32 a^4 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(152*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (122*a^4*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (32*a^4*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])","A",19,6,23,0.2609,1,"{3238, 3791, 3769, 3771, 2639, 2641}"
317,1,164,0,0.1676669,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x]),x]","-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{5 \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 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{5 \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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{3238, 3818, 3787, 3768, 3771, 2639, 2641}"
318,1,136,0,0.1542836,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x]),x]","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{\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 \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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{\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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,7,23,0.3043,1,"{3238, 3818, 3787, 3771, 2641, 3768, 2639}"
319,1,110,0,0.1443569,"\int \frac{\sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x]),x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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,"(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{3238, 3818, 3787, 3771, 2639, 2641}"
320,1,110,0,0.1388344,"\int \frac{1}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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,"-((Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{3238, 3820, 3787, 3771, 2639, 2641}"
321,1,112,0,0.140751,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{\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 \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{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{\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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{3238, 3819, 3787, 3771, 2639, 2641}"
322,1,140,0,0.1615636,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{5 \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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{5 \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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","A",8,7,23,0.3043,1,"{3238, 3819, 3787, 3769, 3771, 2641, 2639}"
323,1,168,0,0.1701167,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)),x]","-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 \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{21 \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{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 \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{21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(21*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (7*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{3238, 3819, 3787, 3769, 3771, 2639, 2641}"
324,1,202,0,0.2730337,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{7 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{10 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{7 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{10 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(7*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (7*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (10*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - (7*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",10,8,23,0.3478,1,"{3238, 3816, 4019, 3787, 3768, 3771, 2639, 2641}"
325,1,176,0,0.2508731,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{5 \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 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{5 \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 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,8,23,0.3478,1,"{3238, 3816, 4019, 3787, 3771, 2641, 3768, 2639}"
326,1,149,0,0.2399155,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^2,x]","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{3238, 3816, 4019, 3787, 3771, 2639, 2641}"
327,1,77,0,0.0969191,"\int \frac{1}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",5,5,23,0.2174,1,"{3238, 3815, 21, 3771, 2641}"
328,1,149,0,0.2379795,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-((Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{3238, 3817, 4019, 3787, 3771, 2639, 2641}"
329,1,152,0,0.2404536,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","-\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 \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 \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{\sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}","-\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 \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 \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{\sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{3238, 3817, 4020, 3787, 3771, 2639, 2641}"
330,1,178,0,0.2636039,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)),x]","\frac{10 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{7 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 \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{\sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}","\frac{10 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{7 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 \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{\sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(-7*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",9,8,23,0.3478,1,"{3238, 3817, 4020, 3787, 3769, 3771, 2641, 2639}"
331,1,200,0,0.2805137,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)),x]","-\frac{3 \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{56 \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{\sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}","-\frac{3 \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{56 \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{\sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(56*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (56*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(a^2*d*Sqrt[Sec[c + d*x]]) - (3*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",10,8,23,0.3478,1,"{3238, 3817, 4020, 3787, 3769, 3771, 2639, 2641}"
332,1,221,0,0.3721458,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3,x]","-\frac{13 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{13 \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 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{8 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}","-\frac{13 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{13 \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 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{8 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-49*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",10,8,23,0.3478,1,"{3238, 3816, 4019, 3787, 3771, 2641, 3768, 2639}"
333,1,195,0,0.3602185,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^3,x]","-\frac{9 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}","-\frac{9 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}",1,"(9*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (9*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{3238, 3816, 4019, 3787, 3771, 2639, 2641}"
334,1,195,0,0.3503347,"\int \frac{1}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3238, 3816, 4019, 4020, 3787, 3771, 2639, 2641}"
335,1,195,0,0.3450857,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"-(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3238, 3815, 4019, 4020, 3787, 3771, 2639, 2641}"
336,1,195,0,0.3564664,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 \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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 \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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2}",1,"(-9*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3238, 3817, 4019, 4020, 3787, 3771, 2639, 2641}"
337,1,195,0,0.3588686,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{13 \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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{8 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}","-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{13 \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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{8 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(49*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{3238, 3817, 4020, 3787, 3771, 2639, 2641}"
338,1,221,0,0.381186,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)),x]","\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 \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 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}","\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 \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 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(-119*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",10,8,23,0.3478,1,"{3238, 3817, 4020, 3787, 3769, 3771, 2641, 2639}"
339,1,153,0,0.2824984,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(32*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])","A",5,3,25,0.1200,1,"{4222, 2772, 2771}"
340,1,115,0,0.2210302,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(16*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",4,3,25,0.1200,1,"{4222, 2772, 2771}"
341,1,77,0,0.1590087,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(4*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{4222, 2772, 2771}"
342,1,36,0,0.1030334,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",2,2,25,0.08000,1,"{4222, 2771}"
343,1,57,0,0.1084214,"\int \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{a} \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 \sqrt{a} \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]*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",3,3,25,0.1200,1,"{4222, 2774, 216}"
344,1,92,0,0.1631163,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{\sqrt{a} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a]*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*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",4,4,25,0.1600,1,"{4222, 2770, 2774, 216}"
345,1,136,0,0.2266859,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Cos[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{a \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \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{3 a \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{a \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \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{3 a \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(3*Sqrt[a]*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*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (3*a*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,4,25,0.1600,1,"{4222, 2770, 2774, 216}"
346,1,161,0,0.3080214,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}",1,"(208*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])","A",6,5,25,0.2000,1,"{4222, 2762, 21, 2772, 2771}"
347,1,121,0,0.2385596,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{6 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{6 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(12*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2762, 21, 2772, 2771}"
348,1,81,0,0.174701,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(10*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{4222, 2762, 21, 2771}"
349,1,96,0,0.1863782,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2),x]","\frac{2 a^{3/2} \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^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^{3/2} \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^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*a^(3/2)*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2762, 21, 2774, 216}"
350,1,95,0,0.177563,"\int (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]],x]","\frac{3 a^{3/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{3 a^{3/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(3*a^(3/2)*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*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2763, 21, 2774, 216}"
351,1,140,0,0.238039,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \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{7 a^2 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{a^2 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{7 a^{3/2} \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{7 a^2 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(7*a^(3/2)*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*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (7*a^2*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2763, 21, 2770, 2774, 216}"
352,1,180,0,0.3053513,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{11 a^2 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \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{11 a^2 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{11 a^2 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{11 a^{3/2} \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{11 a^2 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(11*a^(3/2)*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*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (11*a^2*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (11*a^2*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2763, 21, 2770, 2774, 216}"
353,1,201,0,0.4137588,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{38 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{38 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}",1,"(1168*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",6,5,25,0.2000,1,"{4222, 2762, 2980, 2772, 2771}"
354,1,161,0,0.3467852,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{46 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{92 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{46 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{92 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}",1,"(92*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",5,5,25,0.2000,1,"{4222, 2762, 2980, 2772, 2771}"
355,1,121,0,0.2851064,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2),x]","\frac{22 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{86 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}","\frac{22 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{86 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(86*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (22*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",4,4,25,0.1600,1,"{4222, 2762, 2980, 2771}"
356,1,138,0,0.2876456,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \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{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \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{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a^(5/2)*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 + (14*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{4222, 2762, 2980, 2774, 216}"
357,1,134,0,0.2822886,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2),x]","\frac{5 a^{5/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}","\frac{5 a^{5/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}",1,"(5*a^(5/2)*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*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,25,0.2000,1,"{4222, 2762, 2981, 2774, 216}"
358,1,140,0,0.2922557,"\int (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]],x]","\frac{19 a^{5/2} \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{9 a^3 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}","\frac{19 a^{5/2} \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{9 a^3 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}",1,"(19*a^(5/2)*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) + (9*a^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2763, 2981, 2774, 216}"
359,1,180,0,0.3605387,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{13 a^3 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{25 a^{5/2} \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{25 a^3 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{13 a^3 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{25 a^{5/2} \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{25 a^3 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(25*a^(5/2)*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) + (13*a^3*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (25*a^3*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2763, 2981, 2770, 2774, 216}"
360,1,220,0,0.4185722,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{163 a^3 \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{163 a^{5/2} \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{163 a^3 \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","\frac{163 a^3 \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{163 a^{5/2} \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{163 a^3 \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(163*a^(5/2)*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) + (17*a^3*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (163*a^3*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (163*a^3*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2763, 2981, 2770, 2774, 216}"
361,1,154,0,0.2841522,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(7/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{\cos (c+d x)+1}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{\cos (c+d x)+1}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\cos (c+d x)+1}}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[1 + Cos[c + d*x]])","A",7,6,23,0.2609,1,"{4222, 2779, 2984, 12, 2781, 216}"
362,1,118,0,0.2014229,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\cos (c+d x)+1}}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]])","A",6,6,23,0.2609,1,"{4222, 2779, 2984, 12, 2781, 216}"
363,1,82,0,0.1188155,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])","A",4,4,23,0.1739,1,"{4222, 2779, 2781, 216}"
364,1,47,0,0.0787229,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[1 + Cos[c + d*x]],x]","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d","A",3,3,23,0.1304,1,"{4222, 2781, 216}"
365,1,94,0,0.1538357,"\int \frac{1}{\sqrt{1+\cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[1 + Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d","A",6,5,23,0.2174,1,"{4222, 2777, 2774, 216, 2781}"
366,1,125,0,0.2265022,"\int \frac{1}{\sqrt{1+\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)+1} \sqrt{\sec (c+d x)}}","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)+1} \sqrt{\sec (c+d x)}}",1,"(Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + Sin[c + d*x]/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,6,23,0.2609,1,"{4222, 2778, 2982, 2781, 216, 2774}"
367,1,189,0,0.4210999,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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]*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)) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2779, 2984, 12, 2782, 205}"
368,1,151,0,0.2913829,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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]*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2779, 2984, 12, 2782, 205}"
369,1,113,0,0.1796318,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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]*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2779, 12, 2782, 205}"
370,1,76,0,0.1151593,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\sqrt{2} \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{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*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",3,3,25,0.1200,1,"{4222, 2782, 205}"
371,1,135,0,0.2518636,"\int \frac{1}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{2 \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} \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 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*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]*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",6,6,25,0.2400,1,"{4222, 2777, 2774, 216, 2782, 205}"
372,1,168,0,0.3857063,"\int \frac{1}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","-\frac{\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{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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{\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{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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,"-((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]*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) + Sin[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",7,7,25,0.2800,1,"{4222, 2778, 2982, 2782, 205, 2774, 216}"
373,1,197,0,0.510614,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{11 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}","\frac{11 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"(11*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (7*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2766, 2984, 12, 2782, 205}"
374,1,157,0,0.3528622,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{7 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}","-\frac{7 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(-7*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) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2766, 2984, 12, 2782, 205}"
375,1,117,0,0.2193099,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{3 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{3 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(3*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) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2766, 12, 2782, 205}"
376,1,117,0,0.2157507,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{\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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(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) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2764, 12, 2782, 205}"
377,1,174,0,0.3960892,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{2 \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 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{2 \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 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*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*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) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,7,25,0.2800,1,"{4222, 2765, 2982, 2782, 205, 2774, 216}"
378,1,214,0,0.547534,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)),x]","-\frac{3 \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{9 \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{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}","-\frac{3 \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{9 \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{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(-3*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) + (9*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) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",8,8,25,0.3200,1,"{4222, 2765, 2983, 2982, 2782, 205, 2774, 216}"
379,1,237,0,0.640021,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{95 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{95 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{163 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(163*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (95*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,25,0.2800,1,"{4222, 2766, 2978, 2984, 12, 2782, 205}"
380,1,197,0,0.4946832,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{75 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}","\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{75 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-75*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) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])","A",7,7,25,0.2800,1,"{4222, 2766, 2978, 2984, 12, 2782, 205}"
381,1,157,0,0.3517713,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{19 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{19 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(19*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) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (9*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2766, 2978, 12, 2782, 205}"
382,1,157,0,0.3502784,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{5 \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{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{5 \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{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(5*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) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2764, 2978, 12, 2782, 205}"
383,1,157,0,0.3519902,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{3 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{3 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(3*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) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (7*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2765, 2978, 12, 2782, 205}"
384,1,214,0,0.528317,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{2 \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{43 \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{\sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}","\frac{2 \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{43 \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{\sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*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*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) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,8,25,0.3200,1,"{4222, 2765, 2977, 2982, 2782, 205, 2774, 216}"
385,1,254,0,0.6664363,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)),x]","-\frac{5 \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{35 \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{115 \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{15 \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}","-\frac{5 \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{35 \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{115 \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{15 \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(-5*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) + (115*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) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (15*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{4222, 2765, 2977, 2983, 2982, 2782, 205, 2774, 216}"
386,1,277,0,0.7803248,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{193 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{109 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{629 \sin (c+d x) \sqrt{\sec (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{1015 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}","\frac{193 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{109 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{629 \sin (c+d x) \sqrt{\sec (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{1015 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(1015*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) - (629*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (23*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (109*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (193*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",9,7,25,0.2800,1,"{4222, 2766, 2978, 2984, 12, 2782, 205}"
387,1,237,0,0.6306047,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{691 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{363 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}","\frac{691 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{363 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(-363*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) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (199*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (691*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])","A",8,7,25,0.2800,1,"{4222, 2766, 2978, 2984, 12, 2782, 205}"
388,1,197,0,0.4805724,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(7/2),x]","-\frac{103 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{63 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","-\frac{103 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{63 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(63*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) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (103*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2766, 2978, 12, 2782, 205}"
389,1,197,0,0.4810764,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]),x]","-\frac{5 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{13 \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{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","-\frac{5 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{13 \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{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(13*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) + Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2764, 2978, 12, 2782, 205}"
390,1,197,0,0.4969763,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)),x]","\frac{17 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{7 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","\frac{17 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{7 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(7*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) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (17*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4222, 2765, 2978, 12, 2782, 205}"
391,1,197,0,0.4854403,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)),x]","\frac{67 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{5 \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{\sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}","\frac{67 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{5 \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{\sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(5*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) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (13*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (67*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",7,7,25,0.2800,1,"{4222, 2765, 2977, 2978, 12, 2782, 205}"
392,1,254,0,0.668365,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]","\frac{2 \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{49 \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{177 \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{17 \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","\frac{2 \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{49 \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{177 \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{17 \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(2*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*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) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) - (17*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (49*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",9,8,25,0.3200,1,"{4222, 2765, 2977, 2982, 2782, 205, 2774, 216}"
393,1,294,0,0.8169351,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(9/2)),x]","-\frac{259 \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{7 \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{189 \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{637 \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{7 \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}","-\frac{259 \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{7 \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{189 \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{637 \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{7 \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(-7*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) + (637*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) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) - (7*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (259*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (189*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",10,9,25,0.3600,1,"{4222, 2765, 2977, 2983, 2982, 2782, 205, 2774, 216}"
394,1,237,0,0.6313944,"\int \frac{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)),x]","\frac{73 \sin (c+d x)}{1024 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x)}{256 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{45 \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)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x)}{32 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}","\frac{73 \sin (c+d x)}{1024 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x)}{256 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{45 \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)}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x)}{32 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(45*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]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + (33*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (73*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,7,25,0.2800,1,"{4222, 2765, 2977, 2978, 12, 2782, 205}"
395,1,237,0,0.6361863,"\int \frac{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]","\frac{853 \sin (c+d x)}{3072 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x)}{768 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{35 \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)}{1024 \sqrt{2} a^{9/2} d}-\frac{19 \sin (c+d x)}{96 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}","\frac{853 \sin (c+d x)}{3072 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x)}{768 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{35 \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)}{1024 \sqrt{2} a^{9/2} d}-\frac{19 \sin (c+d x)}{96 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}",1,"(35*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]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)) - (19*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (187*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (853*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])","A",8,7,25,0.2800,1,"{4222, 2765, 2977, 2978, 12, 2782, 205}"
396,1,38,0,0.1182114,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{4}}(c+d x) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/4),x]","\frac{4 a^2 \sin (c+d x) \sqrt[4]{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}","\frac{4 a^2 \sin (c+d x) \sqrt[4]{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(4*a^2*Sec[c + d*x]^(1/4)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{4222, 2762, 8}"
397,1,302,0,0.5314882,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^4,x]","-\frac{a^4 \left(8 m^2+40 m+35\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{4 a^4 (2 m+5) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^4 \left(4 m^2+29 m+55\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{\sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2 \cos ^{m+1}(c+d x)}{d (m+4)}+\frac{2 (m+5) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right) \cos ^{m+1}(c+d x)}{d (m+3) (m+4)}","-\frac{a^4 \left(8 m^2+40 m+35\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{4 a^4 (2 m+5) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^4 \left(4 m^2+29 m+55\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{\sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2 \cos ^{m+1}(c+d x)}{d (m+4)}+\frac{2 (m+5) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right) \cos ^{m+1}(c+d x)}{d (m+3) (m+4)}",1,"(a^4*(55 + 29*m + 4*m^2)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)*(4 + m)) + (Cos[c + d*x]^(1 + m)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) + (2*(5 + m)*Cos[c + d*x]^(1 + m)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)*(4 + m)) - (a^4*(35 + 40*m + 8*m^2)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (4*a^4*(5 + 2*m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",7,6,21,0.2857,1,"{2763, 2976, 2968, 3023, 2748, 2643}"
398,1,232,0,0.3077411,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^3,x]","-\frac{a^3 (4 m+5) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{a^3 (4 m+11) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^3 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{\sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right) \cos ^{m+1}(c+d x)}{d (m+3)}","-\frac{a^3 (4 m+5) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{a^3 (4 m+11) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^3 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{\sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right) \cos ^{m+1}(c+d x)}{d (m+3)}",1,"(a^3*(7 + 2*m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (Cos[c + d*x]^(1 + m)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)) - (a^3*(5 + 4*m)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (a^3*(11 + 4*m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",6,5,21,0.2381,1,"{2763, 2968, 3023, 2748, 2643}"
399,1,173,0,0.1391025,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^2,x]","-\frac{a^2 (2 m+3) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}","-\frac{a^2 (2 m+3) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}",1,"(a^2*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) - (a^2*(3 + 2*m)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a^2*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",4,3,21,0.1429,1,"{2763, 2748, 2643}"
400,1,131,0,0.0637415,"\int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(a + a*Cos[c + d*x]),x]","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{a \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{a \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"-((a*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (a*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",3,2,19,0.1053,1,"{2748, 2643}"
401,1,156,0,0.1258148,"\int \frac{\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx","Int[Cos[c + d*x]^m/(a + a*Cos[c + d*x]),x]","-\frac{\sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\cos ^2(c+d x)\right)}{a d \sqrt{\sin ^2(c+d x)}}+\frac{m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{a d (m+1) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^m(c+d x)}{d (a \cos (c+d x)+a)}","-\frac{\sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\cos ^2(c+d x)\right)}{a d \sqrt{\sin ^2(c+d x)}}+\frac{m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{a d (m+1) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^m(c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[c + d*x]^m*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (Cos[c + d*x]^m*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*Sqrt[Sin[c + d*x]^2]) + (m*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*(1 + m)*Sqrt[Sin[c + d*x]^2])","A",4,3,21,0.1429,1,"{2769, 2748, 2643}"
402,1,229,0,0.3047088,"\int \frac{\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2,x]","\frac{(1-2 m) m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (m+1) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}","\frac{(1-2 m) m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (m+1) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*(1 - m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((1 - 2*m)*m*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(3*a^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (2*(1 - m)*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(3*a^2*d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",5,4,21,0.1905,1,"{2766, 2978, 2748, 2643}"
403,1,150,0,0.101661,"\int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 b x}{128}","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 b x}{128}",1,"(35*b*x)/128 + (a*Sin[c + d*x])/d + (35*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*b*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (b*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)","A",8,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
404,1,128,0,0.0857035,"\int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{d}+\frac{b \sin (c+d x)}{d}",1,"(5*a*x)/16 + (b*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (b*Sin[c + d*x]^3)/d + (3*b*Sin[c + d*x]^5)/(5*d) - (b*Sin[c + d*x]^7)/(7*d)","A",7,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
405,1,114,0,0.081893,"\int \cos ^5(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Cos[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 b x}{16}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 b x}{16}",1,"(5*b*x)/16 + (a*Sin[c + d*x])/d + (5*b*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*b*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",7,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
406,1,92,0,0.0639376,"\int \cos ^4(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(3*a*x)/8 + (b*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*b*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^5)/(5*d)","A",6,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
407,1,76,0,0.0586476,"\int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(3*b*x)/8 + (a*Sin[c + d*x])/d + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*Sin[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
408,1,54,0,0.047293,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(a*x)/2 + (b*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (b*Sin[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
409,1,38,0,0.0153514,"\int \cos (c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}","\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"(b*x)/2 + (a*Sin[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,17,0.05882,1,"{2734}"
410,1,15,0,0.0085139,"\int (a+b \cos (c+d x)) \, dx","Int[a + b*Cos[c + d*x],x]","a x+\frac{b \sin (c+d x)}{d}","a x+\frac{b \sin (c+d x)}{d}",1,"a*x + (b*Sin[c + d*x])/d","A",2,1,10,0.1000,1,"{2637}"
411,1,16,0,0.0232678,"\int (a+b \cos (c+d x)) \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+b x","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+b x",1,"b*x + (a*ArcTanh[Sin[c + d*x]])/d","A",2,2,17,0.1176,1,"{2735, 3770}"
412,1,24,0,0.0369524,"\int (a+b \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d","A",4,4,19,0.2105,1,"{2748, 3767, 8, 3770}"
413,1,47,0,0.0489279,"\int (a+b \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,19,0.2632,1,"{2748, 3768, 3770, 3767, 8}"
414,1,63,0,0.0528197,"\int (a+b \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(b*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Tan[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{2748, 3767, 3768, 3770}"
415,1,85,0,0.065147,"\int (a+b \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*Tan[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{2748, 3768, 3770, 3767}"
416,1,101,0,0.0701793,"\int (a+b \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)","A",6,4,19,0.2105,1,"{2748, 3767, 3768, 3770}"
417,1,150,0,0.1061885,"\int \cos ^4(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2,x]","\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+5 b^2\right)+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}","\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+5 b^2\right)+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"((6*a^2 + 5*b^2)*x)/16 + (2*a*b*Sin[c + d*x])/d + ((6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a*b*Sin[c + d*x]^3)/(3*d) + (2*a*b*Sin[c + d*x]^5)/(5*d)","A",7,5,21,0.2381,1,"{2789, 2633, 3014, 2635, 8}"
418,1,111,0,0.1082673,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2,x]","-\frac{\left(a^2+2 b^2\right) \sin ^3(c+d x)}{3 d}+\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \sin ^5(c+d x)}{5 d}","-\frac{\left(a^2+2 b^2\right) \sin ^3(c+d x)}{3 d}+\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \sin ^5(c+d x)}{5 d}",1,"(3*a*b*x)/4 + ((a^2 + b^2)*Sin[c + d*x])/d + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - ((a^2 + 2*b^2)*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^5)/(5*d)","A",7,5,21,0.2381,1,"{2789, 2635, 8, 3013, 373}"
419,1,101,0,0.0920934,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2,x]","\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+3 b^2\right)-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}","\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+3 b^2\right)-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"((4*a^2 + 3*b^2)*x)/8 + (2*a*b*Sin[c + d*x])/d + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*b*Sin[c + d*x]^3)/(3*d)","A",6,5,21,0.2381,1,"{2789, 2633, 3014, 2635, 8}"
420,1,71,0,0.0498053,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(a^2+b^2\right) \sin (c+d x)}{3 d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x","\frac{2 \left(a^2+b^2\right) \sin (c+d x)}{3 d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x",1,"a*b*x + (2*(a^2 + b^2)*Sin[c + d*x])/(3*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + ((a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",2,2,19,0.1053,1,"{2753, 2734}"
421,1,50,0,0.0149788,"\int (a+b \cos (c+d x))^2 \, dx","Int[(a + b*Cos[c + d*x])^2,x]","\frac{1}{2} x \left(2 a^2+b^2\right)+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}","\frac{1}{2} x \left(2 a^2+b^2\right)+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}",1,"((2*a^2 + b^2)*x)/2 + (2*a*b*Sin[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,12,0.08333,1,"{2644}"
422,1,33,0,0.0624424,"\int (a+b \cos (c+d x))^2 \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x],x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a b x+\frac{b^2 \sin (c+d x)}{d}","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a b x+\frac{b^2 \sin (c+d x)}{d}",1,"2*a*b*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (b^2*Sin[c + d*x])/d","A",3,3,19,0.1579,1,"{2746, 2735, 3770}"
423,1,33,0,0.0656476,"\int (a+b \cos (c+d x))^2 \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+b^2 x","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+b^2 x",1,"b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d","A",4,4,21,0.1905,1,"{2789, 3770, 3012, 8}"
424,1,59,0,0.0784044,"\int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3,x]","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}",1,"((a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,21,0.2381,1,"{2789, 3767, 8, 3012, 3770}"
425,1,80,0,0.0890891,"\int (a+b \cos (c+d x))^2 \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4,x]","\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}","\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}",1,"(a*b*ArcTanh[Sin[c + d*x]])/d + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,21,0.2857,1,"{2789, 3768, 3770, 3012, 3767, 8}"
426,1,110,0,0.0950722,"\int (a+b \cos (c+d x))^2 \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^5,x]","\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}","\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}",1,"((3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Tan[c + d*x])/d + ((3*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*b*Tan[c + d*x]^3)/(3*d)","A",6,5,21,0.2381,1,"{2789, 3767, 3012, 3768, 3770}"
427,1,135,0,0.1084728,"\int (a+b \cos (c+d x))^2 \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^6,x]","\frac{\left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}","\frac{\left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}",1,"(3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) + ((4*a^2 + 5*b^2)*Tan[c + d*x])/(5*d) + (3*a*b*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*b*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (a^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*a^2 + 5*b^2)*Tan[c + d*x]^3)/(15*d)","A",7,5,21,0.2381,1,"{2789, 3768, 3770, 3012, 3767}"
428,1,193,0,0.2149768,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(5 a^2+12 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a \left(5 a^2+12 b^2\right) \sin (c+d x)}{5 d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(18 a^2+5 b^2\right)+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}+\frac{13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{30 d}","-\frac{a \left(a^2+6 b^2\right) \sin ^3(c+d x)}{3 d}+\frac{a \left(a^2+3 b^2\right) \sin (c+d x)}{d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9}{8} a^2 b x+\frac{3 a b^2 \sin ^5(c+d x)}{5 d}+\frac{b^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 b^3 x}{16}",1,"(b*(18*a^2 + 5*b^2)*x)/16 + (a*(5*a^2 + 12*b^2)*Sin[c + d*x])/(5*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (13*a*b^2*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a*(5*a^2 + 12*b^2)*Sin[c + d*x]^3)/(15*d)","A",8,6,21,0.2857,1,"{2793, 3023, 2748, 2633, 2635, 8}"
429,1,180,0,0.2207586,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3,x]","-\frac{\left(-52 a^2 b^2+3 a^4-16 b^4\right) \sin (c+d x)}{30 b d}-\frac{\left(3 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}","-\frac{\left(-52 a^2 b^2+3 a^4-16 b^4\right) \sin (c+d x)}{30 b d}-\frac{\left(3 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}",1,"(a*(4*a^2 + 9*b^2)*x)/8 - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*b*d) - (a*(6*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) - ((3*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)","A",4,3,21,0.1429,1,"{2791, 2753, 2734}"
430,1,121,0,0.1164252,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3,x]","\frac{a \left(a^2+4 b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} b x \left(4 a^2+b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{a \left(a^2+4 b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} b x \left(4 a^2+b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(3*b*(4*a^2 + b^2)*x)/8 + (a*(a^2 + 4*b^2)*Sin[c + d*x])/(2*d) + (b*(2*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + ((a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)","A",3,2,19,0.1053,1,"{2753, 2734}"
431,1,90,0,0.0691565,"\int (a+b \cos (c+d x))^3 \, dx","Int[(a + b*Cos[c + d*x])^3,x]","\frac{2 b \left(4 a^2+b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)+\frac{5 a b^2 \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{b \left(3 a^2+b^2\right) \sin (c+d x)}{d}+a^3 x+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(a*(2*a^2 + 3*b^2)*x)/2 + (2*b*(4*a^2 + b^2)*Sin[c + d*x])/(3*d) + (5*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",2,2,12,0.1667,1,"{2656, 2734}"
432,1,73,0,0.1145798,"\int (a+b \cos (c+d x))^3 \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x],x]","\frac{1}{2} b x \left(6 a^2+b^2\right)+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a b^2 \sin (c+d x)}{2 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))}{2 d}","\frac{1}{2} b x \left(6 a^2+b^2\right)+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a b^2 \sin (c+d x)}{2 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"(b*(6*a^2 + b^2)*x)/2 + (a^3*ArcTanh[Sin[c + d*x]])/d + (5*a*b^2*Sin[c + d*x])/(2*d) + (b^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)","A",4,4,19,0.2105,1,"{2793, 3023, 2735, 3770}"
433,1,68,0,0.1224143,"\int (a+b \cos (c+d x))^3 \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2,x]","-\frac{b \left(a^2-b^2\right) \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))}{d}+3 a b^2 x","-\frac{b \left(a^2-b^2\right) \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))}{d}+3 a b^2 x",1,"3*a*b^2*x + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (b*(a^2 - b^2)*Sin[c + d*x])/d + (a^2*(a + b*Cos[c + d*x])*Tan[c + d*x])/d","A",4,4,21,0.1905,1,"{2792, 3023, 2735, 3770}"
434,1,79,0,0.134308,"\int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3,x]","\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a^2 b \tan (c+d x)}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}+b^3 x","\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a^2 b \tan (c+d x)}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}+b^3 x",1,"b^3*x + (a*(a^2 + 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^2*b*Tan[c + d*x])/(2*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",4,4,21,0.1905,1,"{2792, 3021, 2735, 3770}"
435,1,109,0,0.1821056,"\int (a+b \cos (c+d x))^3 \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4,x]","\frac{a \left(2 a^2+9 b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))}{3 d}","\frac{a \left(2 a^2+9 b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))}{3 d}",1,"(b*(3*a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2 + 9*b^2)*Tan[c + d*x])/(3*d) + (7*a^2*b*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,21,0.2857,1,"{2792, 3021, 2748, 3767, 8, 3770}"
436,1,133,0,0.2034755,"\int (a+b \cos (c+d x))^3 \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5,x]","\frac{b \left(2 a^2+b^2\right) \tan (c+d x)}{d}+\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \left(a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{3 a^2 b \tan (c+d x) \sec ^2(c+d x)}{4 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))}{4 d}","\frac{b \left(2 a^2+b^2\right) \tan (c+d x)}{d}+\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \left(a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{3 a^2 b \tan (c+d x) \sec ^2(c+d x)}{4 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))}{4 d}",1,"(3*a*(a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(2*a^2 + b^2)*Tan[c + d*x])/d + (3*a*(a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a^2*b*Sec[c + d*x]^2*Tan[c + d*x])/(4*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,21,0.3333,1,"{2792, 3021, 2748, 3768, 3770, 3767, 8}"
437,1,169,0,0.2166051,"\int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^6,x]","\frac{a \left(4 a^2+15 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \tan (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(9 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}","\frac{a \left(4 a^2+15 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \tan (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(9 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}",1,"(b*(9*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(4*a^2 + 15*b^2)*Tan[c + d*x])/(5*d) + (b*(9*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (11*a^2*b*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(4*a^2 + 15*b^2)*Tan[c + d*x]^3)/(15*d)","A",7,6,21,0.2857,1,"{2792, 3021, 2748, 3767, 3768, 3770}"
438,1,247,0,0.4008152,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]","-\frac{\left(168 a^2 b^2+35 a^4+24 b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(168 a^2 b^2+35 a^4+24 b^4\right) \sin (c+d x)}{35 d}+\frac{b^2 \left(37 a^2+6 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(6 a^2+5 b^2\right)+\frac{8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d}","-\frac{\left(168 a^2 b^2+35 a^4+24 b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(168 a^2 b^2+35 a^4+24 b^4\right) \sin (c+d x)}{35 d}+\frac{b^2 \left(37 a^2+6 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(6 a^2+5 b^2\right)+\frac{8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(a*b*(6*a^2 + 5*b^2)*x)/4 + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(35*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(37*a^2 + 6*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (8*a*b^3*Cos[c + d*x]^5*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x]^3)/(105*d)","A",9,7,21,0.3333,1,"{2793, 3033, 3023, 2748, 2633, 2635, 8}"
439,1,235,0,0.3202155,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4,x]","-\frac{a \left(-121 a^2 b^2+4 a^4-128 b^4\right) \sin (c+d x)}{60 b d}-\frac{\left(4 a^2-25 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left(-178 a^2 b^2+8 a^4-75 b^4\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(36 a^2 b^2+8 a^4+5 b^4\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}","-\frac{a \left(-121 a^2 b^2+4 a^4-128 b^4\right) \sin (c+d x)}{60 b d}-\frac{\left(4 a^2-25 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{\left(-178 a^2 b^2+8 a^4-75 b^4\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(36 a^2 b^2+8 a^4+5 b^4\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}",1,"((8*a^4 + 36*a^2*b^2 + 5*b^4)*x)/16 - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Sin[c + d*x])/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + ((a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)","A",5,3,21,0.1429,1,"{2791, 2753, 2734}"
440,1,170,0,0.2037509,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4,x]","\frac{2 \left(28 a^2 b^2+3 a^4+4 b^4\right) \sin (c+d x)}{15 d}+\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{1}{2} a b x \left(4 a^2+3 b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}","\frac{2 \left(28 a^2 b^2+3 a^4+4 b^4\right) \sin (c+d x)}{15 d}+\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{1}{2} a b x \left(4 a^2+3 b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"(a*b*(4*a^2 + 3*b^2)*x)/2 + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(15*d) + (a*b*(6*a^2 + 29*b^2)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*a^2 + 4*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",4,2,19,0.1053,1,"{2753, 2734}"
441,1,137,0,0.1468854,"\int (a+b \cos (c+d x))^4 \, dx","Int[(a + b*Cos[c + d*x])^4,x]","\frac{a b \left(19 a^2+16 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2+8 a^4+3 b^4\right)+\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{7 a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}","\frac{a b \left(19 a^2+16 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(24 a^2 b^2+8 a^4+3 b^4\right)+\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{7 a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}",1,"((8*a^4 + 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 + 16*b^2)*Sin[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (7*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",3,3,12,0.2500,1,"{2656, 2753, 2734}"
442,1,107,0,0.2276403,"\int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x],x]","\frac{b^2 \left(17 a^2+2 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(2 a^2+b^2\right)+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}","\frac{b^2 \left(17 a^2+2 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(2 a^2+b^2\right)+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"2*a*b*(2*a^2 + b^2)*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (b^2*(17*a^2 + 2*b^2)*Sin[c + d*x])/(3*d) + (4*a*b^3*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (b^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",5,5,19,0.2632,1,"{2793, 3033, 3023, 2735, 3770}"
443,1,114,0,0.234006,"\int (a+b \cos (c+d x))^4 \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2,x]","-\frac{2 a b \left(a^2-2 b^2\right) \sin (c+d x)}{d}-\frac{b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2+b^2\right)+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))^2}{d}","-\frac{2 a b \left(a^2-2 b^2\right) \sin (c+d x)}{d}-\frac{b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2+b^2\right)+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(b^2*(12*a^2 + b^2)*x)/2 + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(a^2 - 2*b^2)*Sin[c + d*x])/d - (b^2*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d","A",5,5,21,0.2381,1,"{2792, 3033, 3023, 2735, 3770}"
444,1,108,0,0.2521114,"\int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3,x]","-\frac{b^2 \left(a^2-2 b^2\right) \sin (c+d x)}{2 d}+\frac{a^2 \left(a^2+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 b \tan (c+d x)}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x","-\frac{b^2 \left(a^2-2 b^2\right) \sin (c+d x)}{2 d}+\frac{a^2 \left(a^2+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 b \tan (c+d x)}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x",1,"4*a*b^3*x + (a^2*(a^2 + 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a^2 - 2*b^2)*Sin[c + d*x])/(2*d) + (3*a^3*b*Tan[c + d*x])/d + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,21,0.2381,1,"{2792, 3031, 3023, 2735, 3770}"
445,1,115,0,0.2515537,"\int (a+b \cos (c+d x))^4 \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4,x]","\frac{a^2 \left(2 a^2+17 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a^3 b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^4 x","\frac{a^2 \left(2 a^2+17 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a^3 b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^4 x",1,"b^4*x + (2*a*b*(a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2 + 17*b^2)*Tan[c + d*x])/(3*d) + (4*a^3*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",5,5,21,0.2381,1,"{2792, 3031, 3021, 2735, 3770}"
446,1,154,0,0.3358931,"\int (a+b \cos (c+d x))^4 \sec ^5(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5,x]","\frac{4 a b \left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{\left(24 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{5 a^3 b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}","\frac{4 a b \left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{\left(24 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{5 a^3 b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"((3*a^4 + 24*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a*b*(2*a^2 + 3*b^2)*Tan[c + d*x])/(3*d) + (a^2*(3*a^2 + 22*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (5*a^3*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,21,0.3333,1,"{2792, 3031, 3021, 2748, 3767, 8, 3770}"
447,1,188,0,0.3607295,"\int (a+b \cos (c+d x))^4 \sec ^6(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6,x]","\frac{\left(60 a^2 b^2+8 a^4+15 b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \left(4 a^2+27 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{3 a^3 b \tan (c+d x) \sec ^3(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}","\frac{\left(60 a^2 b^2+8 a^4+15 b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \left(4 a^2+27 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{3 a^3 b \tan (c+d x) \sec ^3(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(a*b*(3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + ((8*a^4 + 60*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(15*d) + (a*b*(3*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*(4*a^2 + 27*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (3*a^3*b*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)","A",8,8,21,0.3810,1,"{2792, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
448,1,222,0,0.379992,"\int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^7,x]","\frac{4 a b \left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}","\frac{4 a b \left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}",1,"((5*a^4 + 36*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a*b*(4*a^2 + 5*b^2)*Tan[c + d*x])/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*(5*a^2 + 32*b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (7*a^3*b*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a*b*(4*a^2 + 5*b^2)*Tan[c + d*x]^3)/(15*d)","A",8,7,21,0.3333,1,"{2792, 3031, 3021, 2748, 3767, 3768, 3770}"
449,1,193,0,0.5449326,"\int \frac{\cos ^5(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x]),x]","-\frac{a \left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^5 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(4 a^2 b^2+8 a^4+3 b^4\right)}{8 b^5}-\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}","-\frac{a \left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^4 d}-\frac{2 a^5 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(4 a^2 b^2+8 a^4+3 b^4\right)}{8 b^5}-\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"((8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a^5*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - (a*(3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^4*d) + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",7,6,21,0.2857,1,"{2793, 3049, 3023, 2735, 2659, 205}"
450,1,148,0,0.3265436,"\int \frac{\cos ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x]),x]","\frac{\left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^3 d}+\frac{2 a^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}","\frac{\left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^3 d}+\frac{2 a^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"-(a*(2*a^2 + b^2)*x)/(2*b^4) + (2*a^4*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)","A",6,6,21,0.2857,1,"{2793, 3049, 3023, 2735, 2659, 205}"
451,1,110,0,0.1844059,"\int \frac{\cos ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x]),x]","-\frac{2 a^3 \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\right)}{2 b^3}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{2 a^3 \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\right)}{2 b^3}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*Sin[c + d*x])/(b^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,21,0.2381,1,"{2793, 3023, 2735, 2659, 205}"
452,1,76,0,0.1188074,"\int \frac{\cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x]),x]","\frac{2 a^2 \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 x}{b^2}+\frac{\sin (c+d x)}{b d}","\frac{2 a^2 \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 x}{b^2}+\frac{\sin (c+d x)}{b d}",1,"-((a*x)/b^2) + (2*a^2*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Sin[c + d*x]/(b*d)","A",5,5,21,0.2381,1,"{2746, 12, 2735, 2659, 205}"
453,1,59,0,0.056086,"\int \frac{\cos (c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x]),x]","\frac{x}{b}-\frac{2 a \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{x}{b}-\frac{2 a \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,"x/b - (2*a*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",3,3,19,0.1579,1,"{2735, 2659, 205}"
454,1,49,0,0.0306421,"\int \frac{1}{a+b \cos (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(-1),x]","\frac{2 \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 \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*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",2,2,12,0.1667,1,"{2659, 205}"
455,1,68,0,0.0722405,"\int \frac{\sec (c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 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{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 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*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + ArcTanh[Sin[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{2747, 3770, 2659, 205}"
456,1,85,0,0.1303587,"\int \frac{\sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x]),x]","\frac{2 b^2 \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 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}","\frac{2 b^2 \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 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}",1,"(2*b^2*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (b*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a*d)","A",6,6,21,0.2857,1,"{2802, 12, 2747, 3770, 2659, 205}"
457,1,119,0,0.3242418,"\int \frac{\sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + b*Cos[c + d*x]),x]","-\frac{2 b^3 \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+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{b \tan (c+d x)}{a^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 b^3 \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+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{b \tan (c+d x)}{a^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"(-2*b^3*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)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*Tan[c + d*x])/(a^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",6,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
458,1,157,0,0.5221482,"\int \frac{\sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + b*Cos[c + d*x]),x]","\frac{2 b^4 \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+3 b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}","\frac{2 b^4 \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+3 b^2\right) \tan (c+d x)}{3 a^3 d}-\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"(2*b^4*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(3*a^3*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)","A",7,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
459,1,266,0,0.7229718,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2,x]","\frac{\left(-7 a^2 b^2+12 a^4-2 b^4\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^4 \left(4 a^2-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{a x \left(4 a^2+b^2\right)}{b^5}","\frac{\left(-7 a^2 b^2+12 a^4-2 b^4\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^4 \left(4 a^2-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{a x \left(4 a^2+b^2\right)}{b^5}",1,"-((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,21,0.2857,1,"{2792, 3049, 3023, 2735, 2659, 205}"
460,1,213,0,0.4285657,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^2,x]","-\frac{a \left(3 a^2-2 b^2\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \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\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^2\right)}{2 b^4}","-\frac{a^4 \sin (c+d x)}{b^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \sin (c+d x)}{\sqrt{b^2-a^2} (\cos (c+d x)+1)}\right)}{b^4 d \left(b^2-a^2\right)^{3/2}}+\frac{x \left(6 a^2+b^2\right)}{2 b^4}-\frac{2 a \sin (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^2 d}",1,"((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2792, 3049, 3023, 2735, 2659, 205}"
461,1,155,0,0.2560473,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a x}{b^3}","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a x}{b^3}",1,"(-2*a*x)/b^3 + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + ((2*a^2 - b^2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,21,0.2381,1,"{2792, 3023, 2735, 2659, 205}"
462,1,108,0,0.1427655,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^2,x]","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x}{b^2}","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x}{b^2}",1,"x/b^2 - (2*a*(a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,21,0.1905,1,"{2790, 2735, 2659, 205}"
463,1,85,0,0.0688283,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x])^2,x]","\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}","\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*b*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) + (a*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,19,0.2105,1,"{2754, 12, 2659, 205}"
464,1,86,0,0.0552126,"\int \frac{1}{(a+b \cos (c+d x))^2} \, dx","Int[(a + b*Cos[c + d*x])^(-2),x]","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*a*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",4,4,12,0.3333,1,"{2664, 12, 2659, 205}"
465,1,118,0,0.2185429,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"(-2*b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + ArcTanh[Sin[c + d*x]]/(a^2*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",5,5,19,0.2632,1,"{2802, 3001, 3770, 2659, 205}"
466,1,155,0,0.4070079,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^2,x]","\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2-2 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2-2 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(2*b^2*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - (2*b*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((a^2 - 2*b^2)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
467,1,217,0,0.6814783,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^2,x]","-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \left(2 a^2-3 b^2\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \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^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \left(2 a^2-3 b^2\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \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^3*(4*a^2 - 3*b^2)*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 + 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (b*(2*a^2 - 3*b^2)*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
468,1,270,0,0.9673722,"\int \frac{\sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + b*Cos[c + d*x])^2,x]","\frac{2 b^4 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(7 a^2 b^2+2 a^4-12 b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left(a^2-4 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{2 b^4 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(7 a^2 b^2+2 a^4-12 b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left(a^2-4 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*b^4*(5*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*Sec[c + d*x]*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",8,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
469,1,300,0,0.7832232,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]","-\frac{3 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(-29 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(4 a^2-7 b^2\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{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 b^2+6 a^4+b^4\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^2\right)}{2 b^5}","-\frac{3 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(-29 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(4 a^2-7 b^2\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{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 b^2+6 a^4+b^4\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^2\right)}{2 b^5}",1,"((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - (a^2*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*(4*a^2 - 7*b^2)*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",7,7,21,0.3333,1,"{2792, 3047, 3049, 3023, 2735, 2659, 205}"
470,1,221,0,0.4875194,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^3,x]","\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{3 a^2 \left(-5 a^2 b^2+2 a^4+4 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \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{3 a^3 \left(a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{3 a x}{b^4}","\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{3 a^2 \left(-5 a^2 b^2+2 a^4+4 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \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{3 a^3 \left(a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{3 a x}{b^4}",1,"(-3*a*x)/b^4 + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*a^3*(a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2792, 3031, 3023, 2735, 2659, 205}"
471,1,179,0,0.3035623,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(2 a^2-5 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{x}{b^3}","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(2 a^2-5 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{x}{b^3}",1,"x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*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*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(2*a^2 - 5*b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,21,0.2381,1,"{2792, 3021, 2735, 2659, 205}"
472,1,149,0,0.173072,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"((a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2 - 4*b^2)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,21,0.2381,1,"{2790, 2754, 12, 2659, 205}"
473,1,134,0,0.1231455,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x])^3,x]","\frac{\left(a^2+2 b^2\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 a b \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+2 b^2\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 a b \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}}",1,"(-3*a*b*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) + (a*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 2*b^2)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,4,19,0.2105,1,"{2754, 12, 2659, 205}"
474,1,133,0,0.1095093,"\int \frac{1}{(a+b \cos (c+d x))^3} \, dx","Int[(a + b*Cos[c + d*x])^(-3),x]","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a b \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{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+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{3 a b \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((2*a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (3*a*b*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",5,5,12,0.4167,1,"{2664, 2754, 12, 2659, 205}"
475,1,182,0,0.4561998,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x])^3,x]","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(5 a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(5 a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"-((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^3*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",6,6,19,0.3158,1,"{2802, 3055, 3001, 3770, 2659, 205}"
476,1,232,0,0.7838183,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^3,x]","\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{3 b^2 \left(2 a^2-b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}","\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{3 b^2 \left(2 a^2-b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"(3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*b^2*(2*a^2 - b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
477,1,305,0,1.0776975,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^3,x]","-\frac{b^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(7 a^2-4 b^2\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^2 \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^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(a^2+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(7 a^2-4 b^2\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^2 \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^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*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 + 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b^2*(7*a^2 - 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
478,1,307,0,0.8996818,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4,x]","\frac{\left(-23 a^2 b^2+12 a^4+6 b^4\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-28 a^4 b^2+35 a^2 b^4+8 a^6-20 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(4 a^2-9 b^2\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 \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^3 \left(-11 a^2 b^2+4 a^4+12 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{4 a x}{b^5}","\frac{\left(-23 a^2 b^2+12 a^4+6 b^4\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-28 a^4 b^2+35 a^2 b^4+8 a^6-20 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(4 a^2-9 b^2\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 \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^3 \left(-11 a^2 b^2+4 a^4+12 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{4 a x}{b^5}",1,"(-4*a*x)/b^5 + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*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^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,7,21,0.3333,1,"{2792, 3047, 3031, 3023, 2735, 2659, 205}"
479,1,250,0,0.5711813,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^4,x]","-\frac{a \left(-7 a^4 b^2+8 a^2 b^4+2 a^6-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \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^3 \left(3 a^2-8 b^2\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^2 \left(-28 a^2 b^2+9 a^4+34 b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x}{b^4}","-\frac{a \left(-7 a^4 b^2+8 a^2 b^4+2 a^6-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \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^3 \left(3 a^2-8 b^2\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^2 \left(-28 a^2 b^2+9 a^4+34 b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{x}{b^4}",1,"x/b^4 - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a^3*(3*a^2 - 8*b^2)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2792, 3031, 3021, 2735, 2659, 205}"
480,1,222,0,0.3372239,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^4,x]","-\frac{b \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(-5 a^2 b^2+2 a^4+18 b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}","-\frac{b \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(-5 a^2 b^2+2 a^4+18 b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"-((b*(3*a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(2*a^2 - 7*b^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 - 5*a^2*b^2 + 18*b^4)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,6,21,0.2857,1,"{2792, 3021, 2754, 12, 2659, 205}"
481,1,206,0,0.2810206,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]","\frac{a \left(a^2+4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(a^2-6 b^2\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 b^2+a^4-6 b^4\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}","\frac{a \left(a^2+4 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)^{7/2} (a+b)^{7/2}}-\frac{a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(a^2-6 b^2\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(-10 a^2 b^2+a^4-6 b^4\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"(a*(a^2 + 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2 - 6*b^2)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,5,21,0.2381,1,"{2790, 2754, 12, 2659, 205}"
482,1,192,0,0.2259922,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x])^4,x]","-\frac{b \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\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+3 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","-\frac{b \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\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+3 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"-((b*(4*a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2 + 3*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2 + 13*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,4,19,0.2105,1,"{2754, 12, 2659, 205}"
483,1,184,0,0.2172872,"\int \frac{1}{(a+b \cos (c+d x))^4} \, dx","Int[(a + b*Cos[c + d*x])^(-4),x]","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{5 a b \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{5 a b \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"(a*(2*a^2 + 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (5*a*b*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",6,5,12,0.4167,1,"{2664, 2754, 12, 2659, 205}"
484,1,251,0,0.7875245,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x])^4,x]","-\frac{b \left(-8 a^4 b^2+7 a^2 b^4+8 a^6-2 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-17 a^2 b^2+26 a^4+6 b^4\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^2 \left(8 a^2-3 b^2\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^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}","-\frac{b \left(-8 a^4 b^2+7 a^2 b^4+8 a^6-2 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-17 a^2 b^2+26 a^4+6 b^4\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^2 \left(8 a^2-3 b^2\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^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}",1,"-((b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^4*d) + (b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b^2*(8*a^2 - 3*b^2)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",7,6,19,0.3158,1,"{2802, 3055, 3001, 3770, 2659, 205}"
485,1,308,0,1.270178,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]","\frac{b^2 \left(-35 a^4 b^2+28 a^2 b^4+20 a^6-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-65 a^4 b^2+68 a^2 b^4+6 a^6-24 b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(-11 a^2 b^2+12 a^4+4 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^2 \left(9 a^2-4 b^2\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^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{4 b \tanh ^{-1}(\sin (c+d x))}{a^5 d}","\frac{b^2 \left(-35 a^4 b^2+28 a^2 b^4+20 a^6-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(-65 a^4 b^2+68 a^2 b^4+6 a^6-24 b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(-11 a^2 b^2+12 a^4+4 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^2 \left(9 a^2-4 b^2\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^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{4 b \tanh ^{-1}(\sin (c+d x))}{a^5 d}",1,"(b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b^2*(9*a^2 - 4*b^2)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))","A",8,6,21,0.2857,1,"{2802, 3055, 3001, 3770, 2659, 205}"
486,1,264,0,0.4115665,"\int \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(8 a^2+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(17 a^2 b^2+8 a^4-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^2+19 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \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+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(17 a^2 b^2+8 a^4-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^2+19 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(2*a*(8*a^2 + 19*b^2)*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*(8*a^4 + 17*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)","A",8,8,23,0.3478,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
487,1,207,0,0.28356,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]],x]","\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}","\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}",1,"(-2*(2*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",7,7,23,0.3043,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
488,1,162,0,0.1734448,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2-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 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*a*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)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,21,0.2857,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
489,1,57,0,0.0388896,"\int \sqrt{a+b \cos (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])","A",2,2,14,0.1429,1,"{2655, 2653}"
490,1,118,0,0.2262149,"\int \sqrt{a+b \cos (c+d x)} \sec (c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x],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)}}+\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)}}",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",5,5,21,0.2381,1,"{2803, 2663, 2661, 2807, 2805}"
491,1,197,0,0.4977977,"\int \sqrt{a+b \cos (c+d x)} \sec ^2(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{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{\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{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{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{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{\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{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,"-((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*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (b*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]]) + (Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",9,9,23,0.3913,1,"{2796, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
492,1,262,0,0.7314879,"\int \sqrt{a+b \cos (c+d x)} \sec ^3(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{\left(4 a^2-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{b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{3 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{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{\tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","\frac{\left(4 a^2-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{b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{3 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{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{\tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"-(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*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 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,23,0.4348,1,"{2796, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
493,1,314,0,0.5177432,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(8 a^2+49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(31 a^2 b^2+8 a^4-39 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(33 a^2 b^2+8 a^4+147 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \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+49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(31 a^2 b^2+8 a^4-39 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(33 a^2 b^2+8 a^4+147 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*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*(8*a^4 + 31*a^2*b^2 - 39*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(8*a^2 + 39*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)","A",9,8,23,0.3478,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
494,1,258,0,0.3905608,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(6 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}+\frac{2 \left(-31 a^2 b^2+6 a^4+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(3 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}","-\frac{2 \left(6 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}+\frac{2 \left(-31 a^2 b^2+6 a^4+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(3 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}",1,"(-4*a*(3*a^2 - 41*b^2)*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*(6*a^4 - 31*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)","A",8,7,23,0.3043,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
495,1,199,0,0.2475465,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}","-\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}",1,"(2*(a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(5*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(5*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,21,0.2857,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
496,1,157,0,0.1686855,"\int (a+b \cos (c+d x))^{3/2} \, dx","Int[(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,14,0.4286,1,"{2656, 2752, 2663, 2661, 2655, 2653}"
497,1,179,0,0.314484,"\int (a+b \cos (c+d x))^{3/2} \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x],x]","\frac{2 a^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a 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{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^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a 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{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^2*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",8,8,21,0.3810,1,"{2804, 2655, 2653, 2803, 2663, 2661, 2807, 2805}"
498,1,209,0,0.5418079,"\int (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2,x]","\frac{\left(a^2+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)}{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{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{\left(a^2+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)}{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{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)}}",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^2 + 2*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (3*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",9,9,23,0.3913,1,"{2799, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
499,1,255,0,0.7629611,"\int (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3,x]","\frac{\left(4 a^2+3 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{5 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{7 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{5 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 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}","\frac{\left(4 a^2+3 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{5 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{7 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{5 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 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(-5*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*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 + 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,23,0.4348,1,"{2799, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
500,1,371,0,0.6291577,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(8 a^2+81 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(57 a^2 b^2+8 a^4+135 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left(49 a^4 b^2+78 a^2 b^4+8 a^6-135 b^6\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(51 a^2 b^2+8 a^4+741 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \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+81 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(57 a^2 b^2+8 a^4+135 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left(49 a^4 b^2+78 a^2 b^4+8 a^6-135 b^6\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(51 a^2 b^2+8 a^4+741 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)","A",10,8,23,0.3478,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
501,1,308,0,0.5091873,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 \left(10 a^2-49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}+\frac{4 a \left(-62 a^2 b^2+5 a^4+57 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-279 a^2 b^2+10 a^4-147 b^4\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 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}","-\frac{2 \left(10 a^2-49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}+\frac{4 a \left(-62 a^2 b^2+5 a^4+57 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-279 a^2 b^2+10 a^4-147 b^4\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 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}",1,"(-2*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)","A",9,7,23,0.3043,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
502,1,249,0,0.3595082,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}-\frac{2 \left(2 a^2 b^2+3 a^4-5 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2+29 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}","\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}-\frac{2 \left(2 a^2 b^2+3 a^4-5 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2+29 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}",1,"(2*a*(3*a^2 + 29*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",8,6,21,0.2857,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
503,1,197,0,0.2611721,"\int (a+b \cos (c+d x))^{5/2} \, dx","Int[(a + b*Cos[c + d*x])^(5/2),x]","-\frac{16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{16 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}","-\frac{16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{16 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (16*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",7,7,14,0.5000,1,"{2656, 2753, 2752, 2663, 2661, 2655, 2653}"
504,1,222,0,0.5857093,"\int (a+b \cos (c+d x))^{5/2} \sec (c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]","\frac{2 b \left(2 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a^3 \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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{14 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 \left(2 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a^3 \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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{14 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}}}",1,"(14*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*b*(2*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*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^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,21,0.4286,1,"{2793, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
505,1,222,0,0.5924058,"\int (a+b \cos (c+d x))^{5/2} \sec ^2(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2,x]","\frac{a \left(a^2+4 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{\left(a^2-2 b^2\right) \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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{5 a^2 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{a \left(a^2+4 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{\left(a^2-2 b^2\right) \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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{5 a^2 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"-(((a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (a*(a^2 + 4*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (5*a^2*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d","A",9,9,23,0.3913,1,"{2792, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
506,1,270,0,0.8767031,"\int (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3,x]","\frac{b \left(11 a^2+8 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)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2+15 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^2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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{b \left(11 a^2+8 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)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2+15 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^2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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}}}",1,"(-9*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)]) + (b*(11*a^2 + 8*b^2)*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 + 15*b^2)*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]]) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",10,10,23,0.4348,1,"{2792, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
507,1,323,0,1.169417,"\int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4,x]","\frac{\left(16 a^2+33 b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(16 a^2+59 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+33 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{5 b \left(4 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}","\frac{\left(16 a^2+33 b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(16 a^2+59 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+33 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{5 b \left(4 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}",1,"-((16*a^2 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(16*a^2 + 59*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*(4*a^2 + b^2)*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 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (13*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",11,10,23,0.4348,1,"{2792, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
508,1,246,0,0.3746962,"\int (a+b \cos (c+d x))^{7/2} \, dx","Int[(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 b \left(71 a^2+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-46 a^2 b^2+71 a^4-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{32 a \left(11 a^2+13 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}","\frac{2 b \left(71 a^2+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(-46 a^2 b^2+71 a^4-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{32 a \left(11 a^2+13 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}",1,"(32*a*(11*a^2 + 13*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(71*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (24*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",8,7,14,0.5000,1,"{2656, 2753, 2752, 2663, 2661, 2655, 2653}"
509,1,138,0,0.1851658,"\int \cos ^3(c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{59 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{60 \sqrt{7} d}+\frac{47 E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{\sin (c+d x) \cos (c+d x) (4 \cos (c+d x)+3)^{3/2}}{14 d}-\frac{3 \sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{105 d}","\frac{59 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{60 \sqrt{7} d}+\frac{47 E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{\sin (c+d x) \cos (c+d x) (4 \cos (c+d x)+3)^{3/2}}{14 d}-\frac{3 \sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{105 d}",1,"(47*EllipticE[(c + d*x)/2, 8/7])/(20*Sqrt[7]*d) + (59*EllipticF[(c + d*x)/2, 8/7])/(60*Sqrt[7]*d) + (59*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (3*(3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(70*d) + (Cos[c + d*x]*(3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(14*d)","A",6,6,23,0.2609,1,"{2793, 3023, 2753, 2752, 2661, 2653}"
510,1,105,0,0.137643,"\int \cos ^2(c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[3 + 4*Cos[c + d*x]],x]","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{5 d}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{5 d}",1,"(21*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(20*d) - (Sqrt[7]*EllipticF[(c + d*x)/2, 8/7])/(20*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + ((3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d)","A",5,5,23,0.2174,1,"{2791, 2753, 2752, 2661, 2653}"
511,1,78,0,0.0836391,"\int \cos (c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{6 d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{3 d}","\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{6 d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{3 d}",1,"(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) + (Sqrt[7]*EllipticF[(c + d*x)/2, 8/7])/(6*d) + (2*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2753, 2752, 2661, 2653}"
512,1,23,0,0.011538,"\int \sqrt{3+4 \cos (c+d x)} \, dx","Int[Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}",1,"(2*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d","A",1,1,14,0.07143,1,"{2653}"
513,1,48,0,0.0870154,"\int \sqrt{3+4 \cos (c+d x)} \sec (c+d x) \, dx","Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{8 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{8 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(8*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (6*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",3,3,21,0.1429,1,"{2803, 2661, 2805}"
514,1,95,0,0.2456543,"\int \sqrt{3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx","Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{d}","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{d}",1,"-((Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d) + (3*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/d","A",6,6,23,0.2609,1,"{2796, 3060, 2653, 3002, 2661, 2805}"
515,1,135,0,0.3636177,"\int \sqrt{3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx","Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{5 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{5 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}",1,"-(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(3*d) + (3*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (5*EllipticPi[2, (c + d*x)/2, 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,23,0.3043,1,"{2796, 3055, 3059, 2653, 3002, 2661, 2805}"
516,1,140,0,0.1871827,"\int \sqrt{3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^3,x]","-\frac{59 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{60 \sqrt{7} d}-\frac{47 E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}-\frac{3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{105 d}","-\frac{59 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{60 \sqrt{7} d}-\frac{47 E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}-\frac{3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{105 d}",1,"(-47*EllipticE[(c + Pi + d*x)/2, 8/7])/(20*Sqrt[7]*d) - (59*EllipticF[(c + Pi + d*x)/2, 8/7])/(60*Sqrt[7]*d) + (59*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (3*(3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(70*d) - ((3 - 4*Cos[c + d*x])^(3/2)*Cos[c + d*x]*Sin[c + d*x])/(14*d)","A",6,6,23,0.2609,1,"{2793, 3023, 2753, 2752, 2662, 2654}"
517,1,107,0,0.1379936,"\int \sqrt{3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^2,x]","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{5 d}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{5 d}",1,"(21*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(20*d) - (Sqrt[7]*EllipticF[(c + Pi + d*x)/2, 8/7])/(20*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d)","A",5,5,23,0.2174,1,"{2791, 2753, 2752, 2662, 2654}"
518,1,80,0,0.0838027,"\int \sqrt{3-4 \cos (c+d x)} \cos (c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x],x]","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{6 d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{3 d}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{6 d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{3 d}",1,"-(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(2*d) - (Sqrt[7]*EllipticF[(c + Pi + d*x)/2, 8/7])/(6*d) + (2*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2753, 2752, 2662, 2654}"
519,1,24,0,0.0113837,"\int \sqrt{3-4 \cos (c+d x)} \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}",1,"(2*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d","A",1,1,14,0.07143,1,"{2654}"
520,1,50,0,0.0875074,"\int \sqrt{3-4 \cos (c+d x)} \sec (c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x],x]","-\frac{8 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{6 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}","-\frac{8 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{6 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(-8*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) - (6*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)","A",3,3,21,0.1429,1,"{2803, 2662, 2806}"
521,1,98,0,0.2486461,"\int \sqrt{3-4 \cos (c+d x)} \sec ^2(c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{d}","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{d}",1,"-((Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d) + (3*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/d","A",6,6,23,0.2609,1,"{2796, 3060, 2654, 3002, 2662, 2806}"
522,1,138,0,0.3738318,"\int \sqrt{3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx","Int[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]","-\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{5 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}","-\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{5 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}",1,"(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(3*d) - (3*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) - (5*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(3*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,23,0.3043,1,"{2796, 3055, 3059, 2654, 3002, 2662, 2806}"
523,1,215,0,0.2870466,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 a \left(8 a^2+7 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}","-\frac{2 a \left(8 a^2+7 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*(8*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 + 7*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)","A",7,7,23,0.3043,1,"{2793, 3023, 2752, 2663, 2661, 2655, 2653}"
524,1,165,0,0.1878889,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 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^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}","\frac{2 \left(2 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^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2 + b^2)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,23,0.2609,1,"{2791, 2752, 2663, 2661, 2655, 2653}"
525,1,122,0,0.1078743,"\int \frac{\cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} 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 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} 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*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*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",5,5,21,0.2381,1,"{2752, 2663, 2661, 2655, 2653}"
526,1,57,0,0.0365285,"\int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[1/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",2,2,14,0.1429,1,"{2663, 2661}"
527,1,58,0,0.1260566,"\int \frac{\sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \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 \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*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",2,2,21,0.09524,1,"{2807, 2805}"
528,1,206,0,0.49247,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{\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{\sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \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{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{\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{\sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \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,"-((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)])) + (Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*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]]) + (Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)","A",9,9,23,0.3913,1,"{2802, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
529,1,268,0,0.7128544,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\left(4 a^2+3 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 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 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{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{\tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}","\frac{\left(4 a^2+3 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 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 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{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{\tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"(3*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)]) - (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 + 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (3*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",10,10,23,0.4348,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
530,1,326,0,0.5133772,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 \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^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2-3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{8 a \left(4 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)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^2 b^2+16 a^4-3 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 a^2 \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^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2-3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{8 a \left(4 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)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^2 b^2+16 a^4-3 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(5*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (8*a*(4*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(8*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",8,8,23,0.3478,1,"{2792, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
531,1,257,0,0.34389,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(8 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^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2-5 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 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(8 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^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2-5 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 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*a*(8*a^2 - 5*b^2)*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*(8*a^2 + b^2)*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*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d)","A",7,7,23,0.3043,1,"{2792, 3023, 2752, 2663, 2661, 2655, 2653}"
532,1,186,0,0.2315855,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 \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^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \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^2 \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^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \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*(2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*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^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,23,0.2609,1,"{2790, 2752, 2663, 2661, 2655, 2653}"
533,1,170,0,0.1821833,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 a \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 \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 \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 a \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 \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*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*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*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,21,0.2857,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
534,1,106,0,0.0650911,"\int \frac{1}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Cos[c + d*x])^(-3/2),x]","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}",1,"(2*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*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",4,4,14,0.2857,1,"{2664, 21, 2655, 2653}"
535,1,176,0,0.394094,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b^2 \sin (c+d x)}{a 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)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \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 \sin (c+d x)}{a 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)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \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*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*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*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,21,0.3333,1,"{2802, 3059, 2655, 2653, 12, 2807, 2805}"
536,1,277,0,0.780175,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b \left(a^2-3 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^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{\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^2-3 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^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{\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 - 3*b^2)*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)])) + (Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - (3*b*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 - 3*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + Tan[c + d*x]/(a*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,23,0.4348,1,"{2802, 3056, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
537,1,345,0,1.0817156,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{b^2 \left(7 a^2-15 b^2\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{b \left(7 a^2-15 b^2\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+15 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 b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 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{\tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}","-\frac{b^2 \left(7 a^2-15 b^2\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{b \left(7 a^2-15 b^2\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+15 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 b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 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{\tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}",1,"(b*(7*a^2 - 15*b^2)*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*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 + 15*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b^2*(7*a^2 - 15*b^2)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*b*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])","A",11,10,23,0.4348,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
538,1,436,0,0.8621771,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 \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{8 a^2 \left(2 a^2-3 b^2\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(-71 a^2 b^2+48 a^4+3 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}-\frac{4 a \left(-49 a^2 b^2+32 a^4+7 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(-116 a^2 b^2+128 a^4-17 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-212 a^4 b^2+55 a^2 b^4+128 a^6+9 b^6\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^2 \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{8 a^2 \left(2 a^2-3 b^2\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(-71 a^2 b^2+48 a^4+3 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}-\frac{4 a \left(-49 a^2 b^2+32 a^4+7 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(-116 a^2 b^2+128 a^4-17 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-212 a^4 b^2+55 a^2 b^4+128 a^6+9 b^6\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*(128*a^6 - 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(128*a^4 - 116*a^2*b^2 - 17*b^4)*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^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a^2*(2*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(32*a^4 - 49*a^2*b^2 + 7*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) + (2*(48*a^4 - 71*a^2*b^2 + 3*b^4)*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",9,9,23,0.3913,1,"{2792, 3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
539,1,345,0,0.5700462,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 \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{4 a^3 \left(3 a^2-5 b^2\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-16 a^2 b^2+16 a^4-b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{8 a \left(-7 a^2 b^2+4 a^4+2 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 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 a^2 \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{4 a^3 \left(3 a^2-5 b^2\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-16 a^2 b^2+16 a^4-b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{8 a \left(-7 a^2 b^2+4 a^4+2 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 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*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*(16*a^4 - 16*a^2*b^2 - b^4)*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^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",8,8,23,0.3478,1,"{2792, 3031, 3023, 2752, 2663, 2661, 2655, 2653}"
540,1,281,0,0.378858,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{8 a^2 \left(a^2-2 b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 a \left(8 a^2-9 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 b^2+8 a^4+3 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 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{8 a^2 \left(a^2-2 b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 a \left(8 a^2-9 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-15 a^2 b^2+8 a^4+3 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 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 - 9*b^2)*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*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a^2*(a^2 - 2*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,23,0.3043,1,"{2792, 3021, 2752, 2663, 2661, 2655, 2653}"
541,1,263,0,0.3366768,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2-3 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^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2-3 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^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-4*a*(a^2 - 3*b^2)*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*(2*a^2 - 3*b^2)*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^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*a*(a^2 - 3*b^2)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,23,0.3043,1,"{2790, 2754, 2752, 2663, 2661, 2655, 2653}"
542,1,243,0,0.2715967,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \left(a^2+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*(a^2 + 3*b^2)*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*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*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2 + 3*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,6,21,0.2857,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
543,1,221,0,0.2317842,"\int \frac{1}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Cos[c + d*x])^(-5/2),x]","-\frac{8 a b \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{8 a b \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*b*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,14,0.5000,1,"{2664, 2754, 2752, 2663, 2661, 2655, 2653}"
544,1,320,0,0.8712303,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b^2 \left(7 a^2-3 b^2\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^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b \left(7 a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \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^2 \left(7 a^2-3 b^2\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^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b \left(7 a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \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*b*(7*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*b*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*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^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",10,10,21,0.4762,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
545,1,380,0,1.0999041,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(-26 a^2 b^2+3 a^4+15 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^2-5 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^2-5 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 b^2+3 a^4+15 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 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{\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}","\frac{b \left(-26 a^2 b^2+3 a^4+15 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^2-5 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^2-5 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 b^2+3 a^4+15 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 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{\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"-((3*a^4 - 26*a^2*b^2 + 15*b^4)*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 - 5*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*b*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 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + Tan[c + d*x]/(a*d*(a + b*Cos[c + d*x])^(3/2))","A",11,11,23,0.4783,1,"{2802, 3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
546,1,282,0,0.3578291,"\int \frac{1}{(a+b \cos (c+d x))^{7/2}} \, dx","Int[(a + b*Cos[c + d*x])^(-7/2),x]","-\frac{2 b \left(23 a^2+9 b^2\right) \sin (c+d x)}{15 d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{16 a b \sin (c+d x)}{15 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 b \sin (c+d x)}{5 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{16 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)}{15 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","-\frac{2 b \left(23 a^2+9 b^2\right) \sin (c+d x)}{15 d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{16 a b \sin (c+d x)}{15 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 b \sin (c+d x)}{5 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{16 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)}{15 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) - (16*a*b*Sin[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(23*a^2 + 9*b^2)*Sin[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])","A",8,7,14,0.5000,1,"{2664, 2754, 2752, 2663, 2661, 2655, 2653}"
547,1,111,0,0.1497131,"\int \frac{\cos ^3(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]],x]","-\frac{23 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}","-\frac{23 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}",1,"(9*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(20*d) - (23*EllipticF[(c + d*x)/2, 8/7])/(20*Sqrt[7]*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d) + (Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d)","A",5,5,23,0.2174,1,"{2793, 3023, 2752, 2661, 2653}"
548,1,78,0,0.1017154,"\int \frac{\cos ^2(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{17 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{4 d}+\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d}","\frac{17 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{4 d}+\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d}",1,"-(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(4*d) + (17*EllipticF[(c + d*x)/2, 8/7])/(12*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","A",4,4,23,0.1739,1,"{2791, 2752, 2661, 2653}"
549,1,51,0,0.0516001,"\int \frac{\cos (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}-\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 \sqrt{7} d}","\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}-\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 \sqrt{7} d}",1,"(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) - (3*EllipticF[(c + d*x)/2, 8/7])/(2*Sqrt[7]*d)","A",3,3,21,0.1429,1,"{2752, 2661, 2653}"
550,1,23,0,0.0121487,"\int \frac{1}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[1/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{2 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1,1,14,0.07143,1,"{2661}"
551,1,24,0,0.0387461,"\int \frac{\sec (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{2 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1,1,21,0.04762,1,"{2805}"
552,1,101,0,0.2561205,"\int \frac{\sec ^2(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}","\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}",1,"-(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(3*d) + EllipticF[(c + d*x)/2, 8/7]/(Sqrt[7]*d) - (4*EllipticPi[2, (c + d*x)/2, 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,23,0.2609,1,"{2802, 3060, 2653, 3002, 2661, 2805}"
553,1,137,0,0.3713093,"\int \frac{\sec ^3(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]],x]","-\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{6 d}","-\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{6 d}",1,"(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(3*d) - EllipticF[(c + d*x)/2, 8/7]/(3*Sqrt[7]*d) + (Sqrt[7]*EllipticPi[2, (c + d*x)/2, 8/7])/(3*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)","A",7,7,23,0.3043,1,"{2802, 3055, 3059, 2653, 3002, 2661, 2805}"
554,1,113,0,0.1499016,"\int \frac{\cos ^3(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{23 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{10 d}","\frac{23 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{10 d}",1,"(-9*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(20*d) + (23*EllipticF[(c + Pi + d*x)/2, 8/7])/(20*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]*Sin[c + d*x])/(10*d)","A",5,5,23,0.2174,1,"{2793, 3023, 2752, 2662, 2654}"
555,1,80,0,0.1021202,"\int \frac{\cos ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{17 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{6 d}","\frac{17 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{6 d}",1,"-(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(4*d) + (17*EllipticF[(c + Pi + d*x)/2, 8/7])/(12*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","A",4,4,23,0.1739,1,"{2791, 2752, 2662, 2654}"
556,1,53,0,0.0501004,"\int \frac{\cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}",1,"-(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(2*d) + (3*EllipticF[(c + Pi + d*x)/2, 8/7])/(2*Sqrt[7]*d)","A",3,3,21,0.1429,1,"{2752, 2662, 2654}"
557,1,24,0,0.0119922,"\int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[1/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{2 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{2 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1,1,14,0.07143,1,"{2662}"
558,1,25,0,0.0385874,"\int \frac{\sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]","-\frac{2 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}","-\frac{2 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(-2*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1,1,21,0.04762,1,"{2806}"
559,1,104,0,0.250712,"\int \frac{\sec ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}",1,"-(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(3*d) + EllipticF[(c + Pi + d*x)/2, 8/7]/(Sqrt[7]*d) - (4*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,23,0.2609,1,"{2802, 3060, 2654, 3002, 2662, 2806}"
560,1,140,0,0.3664548,"\int \frac{\sec ^3(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{6 d}","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{6 d}",1,"-(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(3*d) + EllipticF[(c + Pi + d*x)/2, 8/7]/(3*Sqrt[7]*d) - (Sqrt[7]*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)","A",7,7,23,0.3043,1,"{2802, 3055, 3059, 2654, 3002, 2662, 2806}"
561,1,111,0,0.074807,"\int \cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}","\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(6*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*B*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,4,21,0.1905,1,"{2748, 2635, 2639, 2641}"
562,1,87,0,0.0601566,"\int \cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 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*A*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,4,21,0.1905,1,"{2748, 2635, 2641, 2639}"
563,1,61,0,0.0505281,"\int \sqrt{\cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*A*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2748, 2639, 2635, 2641}"
564,1,35,0,0.0388315,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d + (2*A*EllipticF[(c + d*x)/2, 2])/d","A",3,3,21,0.1429,1,"{2748, 2641, 2639}"
565,1,57,0,0.048375,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/Cos[c + d*x]^(3/2),x]","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*A*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,21,0.1905,1,"{2748, 2636, 2639, 2641}"
566,1,83,0,0.0593403,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/Cos[c + d*x]^(5/2),x]","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*A*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,4,21,0.1905,1,"{2748, 2636, 2641, 2639}"
567,1,111,0,0.0736735,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/Cos[c + d*x]^(7/2),x]","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 A \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 A \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*A*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,4,21,0.1905,1,"{2748, 2636, 2639, 2641}"
568,1,160,0,0.1201218,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(9 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{20 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}","\frac{2 \left(9 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{20 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(2*(9*a^2 + 7*b^2)*EllipticE[(c + d*x)/2, 2])/(15*d) + (20*a*b*EllipticF[(c + d*x)/2, 2])/(21*d) + (20*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",7,5,23,0.2174,1,"{2789, 2635, 2641, 3014, 2639}"
569,1,135,0,0.1037007,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(7 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{12 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}","\frac{2 \left(7 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{12 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(12*a*b*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*a^2 + 5*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",6,5,23,0.2174,1,"{2789, 2635, 2639, 3014, 2641}"
570,1,101,0,0.0904047,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(5 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{2 \left(5 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(5*a^2 + 3*b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a*b*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",5,5,23,0.2174,1,"{2789, 2635, 2641, 3014, 2639}"
571,1,72,0,0.0839678,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{2 \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(4*a*b*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,23,0.1739,1,"{2789, 2639, 3014, 2641}"
572,1,68,0,0.0856865,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(3/2),x]","-\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","-\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*(a^2 - b^2)*EllipticE[(c + d*x)/2, 2])/d + (4*a*b*EllipticF[(c + d*x)/2, 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",4,4,23,0.1739,1,"{2789, 2641, 3012, 2639}"
573,1,95,0,0.0949957,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-4*a*b*EllipticE[(c + d*x)/2, 2])/d + (2*(a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2789, 2636, 2639, 3012, 2641}"
574,1,135,0,0.1062299,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(7/2),x]","-\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a*b*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",6,5,23,0.2174,1,"{2789, 2636, 2641, 3012, 2639}"
575,1,194,0,0.22039,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3,x]","\frac{2 a \left(7 a^2+15 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(27 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac{40 a b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}","\frac{2 a \left(7 a^2+15 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(27 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac{40 a b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}",1,"(2*b*(27*a^2 + 7*b^2)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(7*a^2 + 15*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b^2*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)","A",7,6,23,0.2609,1,"{2793, 3023, 2748, 2635, 2641, 2639}"
576,1,159,0,0.2020961,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3,x]","\frac{2 b \left(21 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}","\frac{2 b \left(21 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(2*a*(5*a^2 + 9*b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(21*a^2 + 5*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b^2*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)","A",6,6,23,0.2609,1,"{2793, 3023, 2748, 2639, 2635, 2641}"
577,1,116,0,0.1765838,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{2 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{6 b \left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}+\frac{8 a b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}","\frac{2 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{6 b \left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}+\frac{8 a b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}",1,"(6*b*(5*a^2 + b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/d + (8*a*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",5,5,23,0.2174,1,"{2793, 3023, 2748, 2641, 2639}"
578,1,124,0,0.1878259,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{2 b \left(9 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{d \sqrt{\cos (c+d x)}}","\frac{2 b \left(9 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(-2*a*(a^2 - 3*b^2)*EllipticE[(c + d*x)/2, 2])/d + (2*b*(9*a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2792, 3023, 2748, 2641, 2639}"
579,1,120,0,0.1896717,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(5/2),x]","\frac{2 a \left(a^2+9 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(3 a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 a \left(a^2+9 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(3 a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}",1,"(-2*b*(3*a^2 - b^2)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(a^2 + 9*b^2)*EllipticF[(c + d*x)/2, 2])/(3*d) + (16*a^2*b*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",5,5,23,0.2174,1,"{2792, 3021, 2748, 2641, 2639}"
580,1,149,0,0.2105592,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(7/2),x]","\frac{2 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{6 a \left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{6 a \left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*a*(a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/d + (8*a^2*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (6*a*(a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",6,6,23,0.2609,1,"{2792, 3021, 2748, 2636, 2639, 2641}"
581,1,194,0,0.2333219,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(9/2),x]","\frac{2 a \left(5 a^2+21 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(9 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(5 a^2+21 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(9 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*b*(9*a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*a^2 + 21*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (32*a^2*b*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*a^2 + 21*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*b*(9*a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",7,6,23,0.2609,1,"{2792, 3021, 2748, 2636, 2641, 2639}"
582,1,112,0,0.3908614,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]","\frac{2 \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 \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 \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 \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(-2*a*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*(3*a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) - (2*a^3*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",6,6,23,0.2609,1,"{2793, 3059, 2639, 3002, 2641, 2805}"
583,1,75,0,0.1625451,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x]),x]","\frac{2 a^2 \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 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}","\frac{2 a^2 \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 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*EllipticF[(c + d*x)/2, 2])/(b^2*d) + (2*a^2*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)","A",5,5,23,0.2174,1,"{2804, 2639, 2803, 2641, 2805}"
584,1,53,0,0.1007645,"\int \frac{\sqrt{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}",1,"(2*EllipticF[(c + d*x)/2, 2])/(b*d) - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)","A",3,3,23,0.1304,1,"{2803, 2641, 2805}"
585,1,29,0,0.0453543,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",1,1,23,0.04348,1,"{2805}"
586,1,77,0,0.237682,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{2 b \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}","-\frac{2 b \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(-2*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*b*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d) + (2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2802, 3059, 2639, 12, 2805}"
587,1,128,0,0.5455994,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{2 b^2 \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^2 \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*b*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*b^2*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d) + (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*b*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])","A",7,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
588,1,245,0,0.7048412,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\left(-16 a^2 b^2+15 a^4-2 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{a^3 \left(5 a^2-7 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{a^2 \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^2-2 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}","\frac{\left(-16 a^2 b^2+15 a^4-2 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{a^3 \left(5 a^2-7 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{a^2 \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^2-2 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}",1,"-((a*(5*a^2 - 4*b^2)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d)) + ((15*a^4 - 16*a^2*b^2 - 2*b^4)*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) - (a^3*(5*a^2 - 7*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) + ((5*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",7,7,23,0.3043,1,"{2792, 3049, 3059, 2639, 3002, 2641, 2805}"
589,1,185,0,0.4586156,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2,x]","-\frac{a \left(3 a^2-4 b^2\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^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{a \left(3 a^2-4 b^2\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^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}-\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((3*a^2 - 2*b^2)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,23,0.2609,1,"{2792, 3059, 2639, 3002, 2641, 2805}"
590,1,163,0,0.3844208,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{a \left(a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{a \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^2-2 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{a \left(a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-((a*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,23,0.2609,1,"{2799, 3059, 2639, 3002, 2641, 2805}"
591,1,148,0,0.3967536,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x])^2,x]","\frac{a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}","\frac{a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"EllipticE[(c + d*x)/2, 2]/((a^2 - b^2)*d) + (a*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b*(a + b)^2*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,23,0.2609,1,"{2796, 3059, 2639, 3002, 2641, 2805}"
592,1,157,0,0.436418,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{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-b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{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-b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"-((b*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d)) - EllipticF[(c + d*x)/2, 2]/((a^2 - b^2)*d) + ((3*a^2 - b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",6,6,23,0.2609,1,"{2802, 3059, 2639, 3002, 2641, 2805}"
593,1,217,0,0.683318,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{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^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","\frac{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^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"-(((2*a^2 - 3*b^2)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d)) + (b*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))","A",7,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
594,1,281,0,0.9953911,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\left(2 a^2-5 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{b \left(4 a^2-5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b^2 \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^2-5 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{b \left(4 a^2-5 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}","\frac{\left(2 a^2-5 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{b \left(4 a^2-5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b^2 \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^2-5 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{b \left(4 a^2-5 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"(b*(4*a^2 - 5*b^2)*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (b*(4*a^2 - 5*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))","A",8,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
595,1,346,0,1.0392251,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(9/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-223 a^4 b^2+128 a^2 b^4+105 a^6+8 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(-65 a^2 b^2+35 a^4+24 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(-86 a^2 b^2+35 a^4+63 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{a^2 \left(7 a^2-13 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-61 a^2 b^2+35 a^4+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}","\frac{\left(-223 a^4 b^2+128 a^2 b^4+105 a^6+8 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(-65 a^2 b^2+35 a^4+24 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(-86 a^2 b^2+35 a^4+63 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{a^2 \left(7 a^2-13 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-61 a^2 b^2+35 a^4+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}",1,"-(a*(35*a^4 - 65*a^2*b^2 + 24*b^4)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) + ((105*a^6 - 223*a^4*b^2 + 128*a^2*b^4 + 8*b^6)*EllipticF[(c + d*x)/2, 2])/(12*b^5*(a^2 - b^2)^2*d) - (a^3*(35*a^4 - 86*a^2*b^2 + 63*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((35*a^4 - 61*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(7*a^2 - 13*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",8,8,23,0.3478,1,"{2792, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
596,1,282,0,0.7744704,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^3,x]","-\frac{3 a \left(-11 a^2 b^2+5 a^4+8 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\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^2 \left(-38 a^2 b^2+15 a^4+35 b^4\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^2 \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^2 \left(5 a^2-11 b^2\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{3 a \left(-11 a^2 b^2+5 a^4+8 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\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^2 \left(-38 a^2 b^2+15 a^4+35 b^4\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^2 \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^2 \left(5 a^2-11 b^2\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,"((15*a^4 - 29*a^2*b^2 + 8*b^4)*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(5*a^4 - 11*a^2*b^2 + 8*b^4)*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) + (a^2*(15*a^4 - 38*a^2*b^2 + 35*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - (a^2*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^2*(5*a^2 - 11*b^2)*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",7,7,23,0.3043,1,"{2792, 3047, 3059, 2639, 3002, 2641, 2805}"
597,1,264,0,0.777197,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-5 a^2 b^2+3 a^4+8 b^4\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{3 a \left(a^2-3 b^2\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{3 a \left(-2 a^2 b^2+a^4+5 b^4\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{a^2 \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{3 a \left(a^2-3 b^2\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{\left(-5 a^2 b^2+3 a^4+8 b^4\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{3 a \left(a^2-3 b^2\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{3 a \left(-2 a^2 b^2+a^4+5 b^4\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{a^2 \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{3 a \left(a^2-3 b^2\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))}",1,"(-3*a*(a^2 - 3*b^2)*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,23,0.3043,1,"{2792, 3055, 3059, 2639, 3002, 2641, 2805}"
598,1,244,0,0.657471,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]","\frac{a \left(a^2-7 b^2\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(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 b^2+a^4-3 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}+\frac{\left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \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{a \left(a^2-7 b^2\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(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 b^2+a^4-3 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}+\frac{\left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \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,"-((a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(4*b*(a^2 - b^2)^2*d) + (a*(a^2 - 7*b^2)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,23,0.3043,1,"{2799, 3055, 3059, 2639, 3002, 2641, 2805}"
599,1,250,0,0.6819218,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x])^3,x]","\frac{3 \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 b^2+3 a^4-b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}-\frac{b \left(5 a^2+b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b \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{3 \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 b^2+3 a^4-b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}-\frac{b \left(5 a^2+b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((5*a^2 + b^2)*EllipticE[(c + d*x)/2, 2])/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))","A",7,7,23,0.3043,1,"{2796, 3055, 3059, 2639, 3002, 2641, 2805}"
600,1,261,0,0.7748501,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","-\frac{\left(7 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-b^2\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{3 \left(-2 a^2 b^2+5 a^4+b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}+\frac{3 b^2 \left(3 a^2-b^2\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^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}","-\frac{\left(7 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-b^2\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{3 \left(-2 a^2 b^2+5 a^4+b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}+\frac{3 b^2 \left(3 a^2-b^2\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^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"(-3*b*(3*a^2 - b^2)*EllipticE[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2 - b^2)*EllipticF[(c + d*x)/2, 2])/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*b^2*(3*a^2 - b^2)*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",7,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
601,1,328,0,1.0709629,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{b \left(11 a^2-5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 b^2+8 a^4+15 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-38 a^2 b^2+35 a^4+15 b^4\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{b^2 \left(11 a^2-5 b^2\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^2 \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(-29 a^2 b^2+8 a^4+15 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^2-5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 b^2+8 a^4+15 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-38 a^2 b^2+35 a^4+15 b^4\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{b^2 \left(11 a^2-5 b^2\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^2 \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(-29 a^2 b^2+8 a^4+15 b^4\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"-((8*a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) + (b*(11*a^2 - 5*b^2)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*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",8,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
602,1,395,0,1.3503645,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(-86 a^2 b^2+63 a^4+35 b^4\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^2 \left(13 a^2-7 b^2\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{b^2 \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(-61 a^2 b^2+8 a^4+35 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 \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}","\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(-86 a^2 b^2+63 a^4+35 b^4\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^2 \left(13 a^2-7 b^2\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{b^2 \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(-61 a^2 b^2+8 a^4+35 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 \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"(b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*EllipticF[(c + d*x)/2, 2])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*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^2*(13*a^2 - 7*b^2)*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",9,7,23,0.3043,1,"{2802, 3055, 3059, 2639, 3002, 2641, 2805}"
603,1,438,0,0.9224004,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b} \left(a^2-4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}","\frac{\sqrt{a+b} \left(a^2-4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}",1,"-((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (Sqrt[a + b]*(a + 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)])/(4*b*d) + (Sqrt[a + b]*(a^2 - 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,25,0.2800,1,"{2821, 3061, 3053, 2809, 2998, 2816, 2994}"
604,1,371,0,0.5748017,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{(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{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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{(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{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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"-(((a - b)*Sqrt[a + b]*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]*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 - (a*Sqrt[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) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2821, 3054, 2809, 12, 2801, 2816, 2994}"
605,1,135,0,0.0711268,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","-\frac{2 \csc (c+d x) \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}","-\frac{2 \csc (c+d x) \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(-2*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + d*x]))/(a + b*Cos[c + d*x])]*(a + b*Cos[c + d*x])*Csc[c + d*x]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]]], -((a - b)/(a + b))])/(Sqrt[a + b]*d)","A",1,1,25,0.04000,1,"{2811}"
606,1,229,0,0.2628566,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 (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}} F\left(\sin ^{-1}\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-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 (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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*(a - b)*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)","A",3,3,25,0.1200,1,"{2795, 2816, 2994}"
607,1,271,0,0.4017274,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(5/2),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)}{3 a^2 d}+\frac{2 \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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}","\frac{2 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 \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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\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}",1,"(2*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*(a - b)*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)])/(3*a*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",4,4,25,0.1600,1,"{2796, 2998, 2816, 2994}"
608,1,329,0,0.6456714,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-2 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+2 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 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \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^2-2 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+2 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 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \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 - 2*b^2)*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 + 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)])/(15*a^2*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))","A",5,5,25,0.2000,1,"{2796, 3055, 2998, 2816, 2994}"
609,1,389,0,0.9235507,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2-4 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(25 a^2+6 a b+8 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(19 a^2+8 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \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^2-4 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(25 a^2+6 a b+8 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(19 a^2+8 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*b*Sqrt[a + b]*(19*a^2 + 8*b^2)*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]*(25*a^2 + 6*a*b + 8*b^2)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*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 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))","A",6,5,25,0.2000,1,"{2796, 3055, 2998, 2816, 2994}"
610,1,508,0,1.2568488,"\int \cos ^{\frac{3}{2}}(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])^(3/2),x]","\frac{\left(3 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 d}","\frac{\left(3 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 d}",1,"-((a - b)*Sqrt[a + b]*(3*a^2 + 16*b^2)*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]*(a + 2*b)*(3*a + 8*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) + (a*Sqrt[a + b]*(a^2 - 12*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((3*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,25,0.3200,1,"{2821, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
611,1,433,0,1.170256,"\int \sqrt{\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])^(3/2),x]","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}",1,"(-5*(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)])/(4*d) + (Sqrt[a + b]*(5*a + 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)])/(4*d) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*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) + (3*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + ((a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",8,8,25,0.3200,1,"{2821, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
612,1,375,0,0.6423341,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\frac{b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d 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{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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d 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{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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"-(((a - b)*b*Sqrt[a + b]*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 - (3*a*Sqrt[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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2821, 3053, 2809, 2998, 2816, 2994}"
613,1,337,0,0.471292,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","-\frac{2 (a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{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 (a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{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 - 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)])/d - (2*(a - 2*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (2*b*Sqrt[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","A",5,5,25,0.2000,1,"{2798, 2809, 2998, 2816, 2994}"
614,1,277,0,0.4350606,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\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-3 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}} F\left(\sin ^{-1}\left(\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{8 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a 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-3 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}} F\left(\sin ^{-1}\left(\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{8 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}",1,"(8*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*(a - 3*b)*(a - b)*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)])/(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",4,4,25,0.1600,1,"{2799, 2998, 2816, 2994}"
615,1,325,0,0.6612316,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(3 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{4 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 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) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}","\frac{2 (a-b) \sqrt{a+b} \left(3 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{4 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 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) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}",1,"(2*(a - b)*Sqrt[a + b]*(3*a^2 + b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d) - (2*(a - b)*(3*a - b)*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)])/(5*a*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2))","A",5,5,25,0.2000,1,"{2799, 3055, 2998, 2816, 2994}"
616,1,387,0,0.9492932,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{4 b (a-b) \sqrt{a+b} \left(41 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)}{105 a^3 d}+\frac{16 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{4 b (a-b) \sqrt{a+b} \left(41 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)}{105 a^3 d}+\frac{16 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(4*(a - b)*b*Sqrt[a + b]*(41*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*(a - b)*Sqrt[a + b]*(25*a^2 - 57*a*b - 6*b^2)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (16*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))","A",6,5,25,0.2000,1,"{2799, 3055, 2998, 2816, 2994}"
617,1,454,0,1.3178858,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(11/2),x]","\frac{8 b \left(22 a^2-b^2\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+3 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(-39 a^2 b+147 a^3-6 a b^2-8 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 b^2+147 a^4+8 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{20 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 b \left(22 a^2-b^2\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+3 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(-39 a^2 b+147 a^3-6 a b^2-8 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 b^2+147 a^4+8 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{20 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 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 + 33*a^2*b^2 + 8*b^4)*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]*(147*a^3 - 39*a^2*b - 6*a*b^2 - 8*b^3)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (20*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) + (8*b*(22*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2))","A",7,5,25,0.2000,1,"{2799, 3055, 2998, 2816, 2994}"
618,1,506,0,1.3558942,"\int \sqrt{\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])^(5/2),x]","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}",1,"-((a - b)*Sqrt[a + b]*(33*a^2 + 16*b^2)*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]*(33*a^2 + 26*a*b + 16*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (5*a*Sqrt[a + b]*(a^2 + 4*b^2)*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) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (13*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b^2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,8,25,0.3200,1,"{2793, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
619,1,443,0,1.0025424,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 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)}{4 d}","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 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)}{4 d}",1,"(-9*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) + (Sqrt[a + b]*(8*a^2 + 9*a*b + 2*b^2)*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]*(15*a^2 + 4*b^2)*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) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,25,0.2800,1,"{2793, 3061, 3053, 2809, 2998, 2816, 2994}"
620,1,445,0,1.0101972,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{5 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)}{d}","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{5 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)}{d}",1,"((a - b)*Sqrt[a + b]*(2*a^2 - b^2)*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^2 - 6*a*b - b^2)*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 - (5*a*b*Sqrt[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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2792, 3061, 3053, 2809, 2998, 2816, 2994}"
621,1,392,0,0.7392289,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \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{14 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \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{14 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}",1,"(14*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) + (2*Sqrt[a + b]*(a^2 - 7*a*b + 9*b^2)*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) - (2*b^2*Sqrt[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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,25,0.2400,1,"{2792, 3053, 2809, 2998, 2816, 2994}"
622,1,338,0,0.7646161,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-8 a b+15 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{22 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} \left(9 a^2-8 a b+15 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{22 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2 + 23*b^2)*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*(a - b)*Sqrt[a + b]*(9*a^2 - 8*a*b + 15*b^2)*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^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (22*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",5,5,25,0.2000,1,"{2792, 3055, 2998, 2816, 2994}"
623,1,387,0,1.0443414,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(9/2),x]","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 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)}{21 a d}+\frac{2 b (a-b) \sqrt{a+b} \left(29 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)}{21 a^2 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 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)}{21 a d}+\frac{2 b (a-b) \sqrt{a+b} \left(29 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)}{21 a^2 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(a - b)*b*Sqrt[a + b]*(29*a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d) + (2*(a - b)*Sqrt[a + b]*(5*a^2 - 24*a*b + 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))","A",6,5,25,0.2000,1,"{2792, 3055, 2998, 2816, 2994}"
624,1,454,0,1.4087546,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(11/2),x]","\frac{2 \left(49 a^2+75 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 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-114 a^2 b+147 a^3+165 a b^2+10 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 b^2+147 a^4-10 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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{38 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(49 a^2+75 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 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-114 a^2 b+147 a^3+165 a b^2+10 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 b^2+147 a^4-10 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^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{38 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(a - b)*Sqrt[a + b]*(147*a^4 + 279*a^2*b^2 - 10*b^4)*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]*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*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^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (38*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 + 75*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2))","A",7,5,25,0.2000,1,"{2792, 3055, 2998, 2816, 2994}"
625,1,522,0,1.7745748,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(13/2),x]","\frac{2 \left(81 a^2+113 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 b \left(229 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(205 a^2 b^2+135 a^4-4 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(57 a^2 b^2-606 a^3 b+135 a^4+6 a b^3+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(51 a^2 b^2+741 a^4+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{46 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 \left(81 a^2+113 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 b \left(229 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(205 a^2 b^2+135 a^4-4 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(57 a^2 b^2-606 a^3 b+135 a^4+6 a b^3+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{2 b (a-b) \sqrt{a+b} \left(51 a^2 b^2+741 a^4+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{46 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(a - b)*b*Sqrt[a + b]*(741*a^4 + 51*a^2*b^2 + 8*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d) + (2*(a - b)*Sqrt[a + b]*(135*a^4 - 606*a^3*b + 57*a^2*b^2 + 6*a*b^3 + 8*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (46*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 + 113*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*b*(229*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a*d*Cos[c + d*x]^(5/2)) + (2*(135*a^4 + 205*a^2*b^2 - 4*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2))","A",8,5,25,0.2000,1,"{2792, 3055, 2998, 2816, 2994}"
626,1,414,0,0.7357784,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\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{(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 \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{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{\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{(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]*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]*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]*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*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])","A",8,8,25,0.3200,1,"{2820, 2809, 3003, 2993, 12, 2801, 2816, 2994}"
627,1,116,0,0.0684566,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \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 \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]*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",1,1,25,0.04000,1,"{2809}"
628,1,109,0,0.0709568,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \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 \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]*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",1,1,25,0.04000,1,"{2816}"
629,1,224,0,0.233901,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \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 (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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*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)","A",3,3,25,0.1200,1,"{2801, 2816, 2994}"
630,1,274,0,0.4036164,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \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+2 b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-4*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(a + 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^2*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))","A",4,4,25,0.1600,1,"{2802, 2998, 2816, 2994}"
631,1,465,0,0.9862287,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b}}+\frac{(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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b}}+\frac{(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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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,"-(((3*a^2 - b^2)*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)) + ((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)])/(b^2*Sqrt[a + b]*d) + (3*a*Sqrt[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^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",7,7,25,0.2800,1,"{2792, 3061, 3053, 2809, 2998, 2816, 2994}"
632,1,387,0,0.5025676,"\int \frac{\cos ^{\frac{3}{2}}(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])^(3/2),x]","-\frac{2 a^2 \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 \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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\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{a+b}}","-\frac{2 a^2 \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 \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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\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{a+b}}",1,"(2*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)])/(b*Sqrt[a + b]*d) - (2*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*Sqrt[a + b]*d) - (2*Sqrt[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) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2797, 2809, 2794, 2795, 2816, 2994}"
633,1,266,0,0.3345652,"\int \frac{\sqrt{\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])^(3/2),x]","\frac{2 a \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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\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 \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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\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*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*Sqrt[a + b]*d) + (2*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*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2794, 2795, 2816, 2994}"
634,1,267,0,0.3876383,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{2 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 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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}","-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 b \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\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 \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"(2*b*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*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2800, 2998, 2816, 2994}"
635,1,285,0,0.4611821,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b^2 \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^2-2 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+2 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^2 \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^2-2 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+2 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 - 2*b^2)*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*(a + 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^2*Sqrt[a + b]*d) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2802, 2998, 2816, 2994}"
636,1,357,0,0.7240714,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b}}+\frac{2 (a+2 b) (a+4 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 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b}}+\frac{2 (a+2 b) (a+4 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*b*(5*a^2 - 8*b^2)*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)*(a + 4*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^2*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 - 4*b^2)*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",5,5,25,0.2000,1,"{2802, 3055, 2998, 2816, 2994}"
637,1,433,0,1.0569566,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (3 a+4 b) \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (3 a+4 b) \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}",1,"(2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d) - (2*(3*a + 4*b)*(a^2 + 4*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 6*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) - (2*b*(3*a^2 - 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))","A",6,5,25,0.2000,1,"{2802, 3055, 2998, 2816, 2994}"
638,1,497,0,1.064423,"\int \frac{\cos ^{\frac{5}{2}}(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])^(5/2),x]","-\frac{2 a^2 \left(3 a^2-7 b^2\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^2 \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(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \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^2 \left(3 a^2-7 b^2\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^2 \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(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \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*(3*a^2 - 7*b^2)*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 - b)*b^2*(a + b)^(3/2)*d) - (2*(3*a^2 + a*b - 6*b^2)*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^2*(a + b)^(3/2)*d) - (2*Sqrt[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^2*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^2*(3*a^2 - 7*b^2)*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",7,7,25,0.2800,1,"{2792, 3051, 2809, 2993, 2998, 2816, 2994}"
639,1,342,0,0.6099342,"\int \frac{\cos ^{\frac{3}{2}}(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])^(5/2),x]","-\frac{8 a b \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 \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 (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 (a-b) (a+b)^{3/2}}+\frac{8 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 (a-b) (a+b)^{3/2}}","-\frac{8 a b \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 \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 (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 (a-b) (a+b)^{3/2}}+\frac{8 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 (a-b) (a+b)^{3/2}}",1,"(8*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)*(a + b)^(3/2)*d) + (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)])/(3*a*(a - b)*(a + b)^(3/2)*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*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",5,5,25,0.2000,1,"{2799, 2993, 2998, 2816, 2994}"
640,1,359,0,0.6416851,"\int \frac{\sqrt{\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])^(5/2),x]","\frac{2 \left(3 a^2+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 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+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-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+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 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+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-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 + b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*(3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^2 + b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2796, 2993, 2998, 2816, 2994}"
641,1,381,0,0.7315844,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2-3 a b-2 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 (a-b) (a+b)^{3/2}}+\frac{4 b \left(3 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2-3 a b-2 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 (a-b) (a+b)^{3/2}}+\frac{4 b \left(3 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(4*b*(3*a^2 - b^2)*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*(3*a^2 - 3*a*b - 2*b^2)*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*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*b*(3*a^2 - b^2)*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",5,5,25,0.2000,1,"{2802, 2993, 2998, 2816, 2994}"
642,1,398,0,0.8130928,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{8 b^2 \left(2 a^2-b^2\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^2 \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(9 a^2 b+3 a^3-6 a b^2-8 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 (a-b) (a+b)^{3/2}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 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{8 b^2 \left(2 a^2-b^2\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^2 \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(9 a^2 b+3 a^3-6 a b^2-8 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 (a-b) (a+b)^{3/2}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 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 - 15*a^2*b^2 + 8*b^4)*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*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*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*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*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",5,5,25,0.2000,1,"{2802, 3055, 2998, 2816, 2994}"
643,1,473,0,1.1732421,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{4 b^2 \left(5 a^2-3 b^2\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^2 \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(-13 a^2 b^2+a^4+8 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 \left(16 a^2 b^2+9 a^3 b+a^4-12 a b^3-16 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 (a-b) (a+b)^{3/2}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2}}","\frac{4 b^2 \left(5 a^2-3 b^2\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^2 \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(-13 a^2 b^2+a^4+8 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 \left(16 a^2 b^2+9 a^3 b+a^4-12 a b^3-16 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 (a-b) (a+b)^{3/2}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2}}",1,"(-8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*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*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*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*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*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 - 13*a^2*b^2 + 8*b^4)*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",6,5,25,0.2000,1,"{2802, 3055, 2998, 2816, 2994}"
644,1,32,0,0.0588819,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{2+3 \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]","\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d}","\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d}",1,"(2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d)","A",1,1,25,0.04000,1,"{2813}"
645,1,25,0,0.0525776,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{-2+3 \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]),x]","\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d}","\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d}",1,"(2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 5])/d","A",1,1,25,0.04000,1,"{2813}"
646,1,56,0,0.1169758,"\int \frac{1}{\sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Int[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{\cos (c+d x)}}","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{\cos (c+d x)}}",1,"(-2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2814, 2813}"
647,1,49,0,0.1103613,"\int \frac{1}{\sqrt{-2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Int[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d \sqrt{\cos (c+d x)}}","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d \sqrt{\cos (c+d x)}}",1,"(-2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/(d*Sqrt[Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2814, 2813}"
648,1,58,0,0.0529605,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d","A",1,1,25,0.04000,1,"{2815}"
649,1,60,0,0.0661629,"\int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Int[1/(Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d)","A",1,1,25,0.04000,1,"{2815}"
650,1,84,0,0.1235994,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d)","A",2,2,25,0.08000,1,"{2817, 2815}"
651,1,82,0,0.1146251,"\int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Int[1/(Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d","A",2,2,25,0.08000,1,"{2817, 2815}"
652,1,54,0,0.1009972,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{2+3 \cos (c+d x)}} \, dx","Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])","A",2,2,27,0.07407,1,"{2814, 2813}"
653,1,47,0,0.1038505,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-2+3 \cos (c+d x)}} \, dx","Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d \sqrt{-\cos (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 5])/(d*Sqrt[-Cos[c + d*x]])","A",2,2,27,0.07407,1,"{2814, 2813}"
654,1,34,0,0.0552737,"\int \frac{1}{\sqrt{2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Int[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","-\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d}","-\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d}",1,"(-2*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 1/5])/(Sqrt[5]*d)","A",1,1,27,0.03704,1,"{2813}"
655,1,27,0,0.0553474,"\int \frac{1}{\sqrt{-2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Int[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","-\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d}","-\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d}",1,"(-2*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/d","A",1,1,27,0.03704,1,"{2813}"
656,1,80,0,0.1047984,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx","Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(2*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/(d*Sqrt[-Cos[c + d*x]])","A",2,2,27,0.07407,1,"{2817, 2815}"
657,1,82,0,0.1059277,"\int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Int[1/(Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(2*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])","A",2,2,27,0.07407,1,"{2817, 2815}"
658,1,62,0,0.05566,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx","Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(-2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d)","A",1,1,27,0.03704,1,"{2815}"
659,1,60,0,0.0556197,"\int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Int[1/(Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d","A",1,1,27,0.03704,1,"{2815}"
660,1,77,0,0.0488477,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2+3 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]],x]","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(-4*Cot[c + d*x]*EllipticPi[5/3, 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]])/(3*d)","A",1,1,25,0.04000,1,"{2809}"
661,1,75,0,0.0590285,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-2+3 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]],x]","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(-4*Cot[c + d*x]*EllipticPi[1/3, 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]])/(3*Sqrt[5]*d)","A",1,1,25,0.04000,1,"{2809}"
662,1,99,0,0.116352,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]],x]","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d \sqrt{-\cos (c+d x)}}","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(-4*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[1/3, 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]])/(3*Sqrt[5]*d*Sqrt[-Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2810, 2809}"
663,1,101,0,0.1024294,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d \sqrt{-\cos (c+d x)}}","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d \sqrt{-\cos (c+d x)}}",1,"(-4*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[5/3, 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]])/(3*d*Sqrt[-Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2810, 2809}"
664,1,73,0,0.0466321,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-3*Cot[c + d*x]*EllipticPi[5/2, 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]])/d","A",1,1,25,0.04000,1,"{2808}"
665,1,75,0,0.049012,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{3-2 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]],x]","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(3*Cot[c + d*x]*EllipticPi[-1/2, 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]])/(Sqrt[5]*d)","A",1,1,25,0.04000,1,"{2808}"
666,1,99,0,0.1004558,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-3+2 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]],x]","\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}","\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(3*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[-1/2, 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]])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2810, 2808}"
667,1,97,0,0.1028067,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-3-2 \cos (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]],x]","-\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}","-\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[5/2, 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]])/(d*Sqrt[-Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2810, 2808}"
668,1,99,0,0.0993077,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{2+3 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]],x]","-\frac{4 \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)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}","-\frac{4 \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)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(-4*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[5/3, 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]])/(3*d)","A",2,2,27,0.07407,1,"{2810, 2809}"
669,1,97,0,0.0984587,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2+3 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]],x]","-\frac{4 \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} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}","-\frac{4 \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} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(-4*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[1/3, 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]])/(3*Sqrt[5]*d)","A",2,2,27,0.07407,1,"{2810, 2809}"
670,1,77,0,0.0520857,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]],x]","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(-4*Cot[c + d*x]*EllipticPi[1/3, 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]])/(3*Sqrt[5]*d)","A",1,1,27,0.03704,1,"{2809}"
671,1,79,0,0.0536822,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d}","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(-4*Cot[c + d*x]*EllipticPi[5/3, 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]])/(3*d)","A",1,1,27,0.03704,1,"{2809}"
672,1,95,0,0.1002335,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]","-\frac{3 \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} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}","-\frac{3 \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} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[5/2, 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]])/d","A",2,2,27,0.07407,1,"{2810, 2808}"
673,1,97,0,0.100714,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{3-2 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]],x]","\frac{3 \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} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","\frac{3 \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} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[-1/2, 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]])/(Sqrt[5]*d)","A",2,2,27,0.07407,1,"{2810, 2808}"
674,1,77,0,0.0532438,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3+2 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]],x]","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(3*Cot[c + d*x]*EllipticPi[-1/2, 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]])/(Sqrt[5]*d)","A",1,1,27,0.03704,1,"{2808}"
675,1,75,0,0.0525886,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3-2 \cos (c+d x)}} \, dx","Int[Sqrt[-Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]],x]","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-3*Cot[c + d*x]*EllipticPi[5/2, 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]])/d","A",1,1,27,0.03704,1,"{2808}"
676,1,176,0,0.1991257,"\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(2/3)/(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}-\frac{b \sin (c+d x) \cos ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos ^2(c+d x)}}","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}-\frac{b \sin (c+d x) \cos ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos ^2(c+d x)}}",1,"-((b*AppellF1[1/2, -1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3))) + (a*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(1/3))","A",5,3,23,0.1304,1,"{2823, 3189, 429}"
677,1,176,0,0.185033,"\int \frac{\sqrt[3]{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^(1/3)/(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos (c+d x)} F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[6]{\cos ^2(c+d x)}}","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos (c+d x)} F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[6]{\cos ^2(c+d x)}}",1,"-((b*AppellF1[1/2, -1/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6))) + (a*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(2/3))","A",5,3,23,0.1304,1,"{2823, 3189, 429}"
678,1,176,0,0.1864234,"\int \frac{1}{\sqrt[3]{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])),x]","\frac{a \sin (c+d x) \cos ^2(c+d x)^{2/3} F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{4}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}","\frac{a \sin (c+d x) \cos ^2(c+d x)^{2/3} F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{4}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}",1,"-((b*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(1/3))) + (a*AppellF1[1/2, 2/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(4/3))","A",5,3,23,0.1304,1,"{2823, 3189, 429}"
679,1,176,0,0.185417,"\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])),x]","\frac{a \sin (c+d x) \cos ^2(c+d x)^{5/6} F_1\left(\frac{1}{2};\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{5}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}","\frac{a \sin (c+d x) \cos ^2(c+d x)^{5/6} F_1\left(\frac{1}{2};\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{5}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}",1,"-((b*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(2/3))) + (a*AppellF1[1/2, 5/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(5/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(5/3))","A",5,3,23,0.1304,1,"{2823, 3189, 429}"
680,0,0,0,0.0535747,"\int \frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",0,0,0,0,-1,"{}"
681,0,0,0,0.0543584,"\int \frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",0,0,0,0,-1,"{}"
682,0,0,0,0.0545425,"\int \frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",0,0,0,0,-1,"{}"
683,0,0,0,0.0549703,"\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",0,0,0,0,-1,"{}"
684,0,0,0,0.0537361,"\int \frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",0,0,0,0,-1,"{}"
685,0,0,0,0.0530685,"\int \frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
686,0,0,0,0.0530306,"\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
687,0,0,0,0.0533637,"\int \frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(4/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][1/(Cos[c + d*x]^(4/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
688,0,0,0,0.0535666,"\int \frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
689,0,0,0,0.0535432,"\int \frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Defer[Int][1/(Cos[c + d*x]^(7/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
690,1,151,0,0.1122955,"\int (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 A \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 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 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 A \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 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,"(-6*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,6,21,0.2857,1,"{3238, 3787, 3768, 3771, 2641, 2639}"
691,1,123,0,0.0962124,"\int (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,6,21,0.2857,1,"{3238, 3787, 3768, 3771, 2639, 2641}"
692,1,97,0,0.0843604,"\int (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,21,0.2857,1,"{3238, 3787, 3771, 2641, 3768, 2639}"
693,1,75,0,0.0757031,"\int (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Int[(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 A \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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d","A",6,5,21,0.2381,1,"{3238, 3787, 3771, 2639, 2641}"
694,1,101,0,0.0867673,"\int \frac{A+B \cos (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/Sqrt[Sec[c + d*x]],x]","\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)}{d}+\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 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 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*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2641, 2639}"
695,1,127,0,0.1033039,"\int \frac{A+B \cos (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2),x]","\frac{2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 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*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2639, 2641}"
696,1,151,0,0.1135,"\int \frac{A+B \cos (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Cos[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 B \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 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{2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 B \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 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}",1,"(6*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*B*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,6,21,0.2857,1,"{3238, 3787, 3769, 3771, 2641, 2639}"
697,1,200,0,0.1663214,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2),x]","\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{12 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{12 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 \left(5 a^2+7 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{12 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{12 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}",1,"(-12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (12*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*a^2 + 7*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",10,7,23,0.3043,1,"{3238, 3788, 3768, 3771, 2639, 4046, 2641}"
698,1,175,0,0.1571202,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2),x]","\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 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 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 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}",1,"(-2*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",9,7,23,0.3043,1,"{3238, 3788, 3768, 3771, 2641, 4046, 2639}"
699,1,135,0,0.1362629,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2),x]","\frac{2 \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}","\frac{2 \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}",1,"(-4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,23,0.3043,1,"{3238, 3788, 3768, 3771, 2639, 4046, 2641}"
700,1,108,0,0.1295557,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2),x]","-\frac{2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 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 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 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}",1,"(-2*(a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,23,0.2609,1,"{3238, 3788, 3771, 2641, 4046, 2639}"
701,1,112,0,0.1324997,"\int (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]],x]","\frac{2 \left(3 a^2+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{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)}{d}+\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2+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{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)}{d}+\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,23,0.2609,1,"{3238, 3788, 3771, 2639, 4045, 2641}"
702,1,141,0,0.1443602,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{2 \left(5 a^2+3 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{4 a b \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 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 b^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(5 a^2+3 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{4 a b \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 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 b^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,23,0.3043,1,"{3238, 3788, 3769, 3771, 2641, 4045, 2639}"
703,1,175,0,0.1641792,"\int \frac{(a+b \cos (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{12 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^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{12 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^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,23,0.3043,1,"{3238, 3788, 3769, 3771, 2639, 4045, 2641}"
704,1,200,0,0.1741458,"\int \frac{(a+b \cos (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2/Sec[c + d*x]^(5/2),x]","\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{20 a b \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{20 a 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{2 b^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{20 a b \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{20 a 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{2 b^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*(9*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (20*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (4*a*b*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(9*a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (20*a*b*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",10,7,23,0.3043,1,"{3238, 3788, 3769, 3771, 2641, 4045, 2639}"
705,1,234,0,0.2689067,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(9 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac{32 a^2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(9 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac{32 a^2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}",1,"(-2*b*(9*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(9*a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*a^2 + 21*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (32*a^2*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a^2*Sec[c + d*x]^(5/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(7*d)","A",10,8,23,0.3478,1,"{3238, 3842, 4047, 3768, 3771, 2641, 4046, 2639}"
706,1,189,0,0.2378386,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2),x]","\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(a^2+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)}{d}-\frac{6 a \left(a^2+5 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{8 a^2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}","\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(a^2+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)}{d}-\frac{6 a \left(a^2+5 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{8 a^2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}",1,"(-6*a*(a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (6*a*(a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^2*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",9,8,23,0.3478,1,"{3238, 3842, 4047, 3768, 3771, 2639, 4046, 2641}"
707,1,160,0,0.2312296,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2),x]","\frac{2 a \left(a^2+9 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 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}+\frac{16 a^2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}","\frac{2 a \left(a^2+9 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 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}+\frac{16 a^2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}",1,"(-2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,23,0.3043,1,"{3238, 3842, 4047, 3771, 2641, 4046, 2639}"
708,1,166,0,0.2285183,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2),x]","\frac{2 a \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 b \left(9 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 b \left(9 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{3 d \sqrt{\sec (c+d x)}}",1,"(-2*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*b*(9*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,23,0.3043,1,"{3238, 3841, 4047, 3771, 2641, 4046, 2639}"
709,1,156,0,0.2229202,"\int (a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]],x]","\frac{2 a \left(a^2+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)}{d}+\frac{6 b \left(5 a^2+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 \sin (c+d x) (a \sec (c+d x)+b)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b^2 \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}","\frac{2 a \left(a^2+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)}{d}+\frac{6 b \left(5 a^2+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 \sin (c+d x) (a \sec (c+d x)+b)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b^2 \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}",1,"(6*b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a*b^2*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",8,7,23,0.3043,1,"{3238, 3841, 4047, 3771, 2639, 4045, 2641}"
710,1,199,0,0.2475324,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(5 a^2+9 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 \sin (c+d x) (a \sec (c+d x)+b)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a b^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(5 a^2+9 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 \sin (c+d x) (a \sec (c+d x)+b)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a b^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*a*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (32*a*b^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",9,8,23,0.3478,1,"{3238, 3841, 4047, 3769, 3771, 2641, 4045, 2639}"
711,1,234,0,0.2768594,"\int \frac{(a+b \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2+15 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(27 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2+15 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(27 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*b*(27*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (40*a*b^2*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*b*(27*a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*a^2 + 15*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",10,8,23,0.3478,1,"{3238, 3841, 4047, 3769, 3771, 2639, 4045, 2641}"
712,1,188,0,0.5468389,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]","\frac{2 b^2 \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 b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 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)}{a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}","\frac{2 b^2 \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 b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 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)}{a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",1,"(2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*b^2*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*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",11,10,23,0.4348,1,"{3238, 3851, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
713,1,117,0,0.2088983,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x]),x]","-\frac{2 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 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","-\frac{2 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 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (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)*d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",7,7,23,0.3043,1,"{3238, 3850, 3768, 3771, 2639, 3849, 2805}"
714,1,49,0,0.1305932,"\int \frac{\sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{2 \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)}{d (a+b)}","\frac{2 \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)}{d (a+b)}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d)","A",3,3,23,0.1304,1,"{3238, 3849, 2805}"
715,1,93,0,0.1870719,"\int \frac{1}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a \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 \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 \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)}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*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",5,5,23,0.2174,1,"{3238, 3848, 2803, 2641, 2805}"
716,1,135,0,0.2358543,"\int \frac{1}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \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 \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^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \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 \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*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*a^2*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",9,8,23,0.3478,1,"{3238, 3852, 3849, 2805, 3787, 3771, 2639, 2641}"
717,1,172,0,0.3910524,"\int \frac{1}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a^3 \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 \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 \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a^3 \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 \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 \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"(-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a^3*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*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])","A",10,9,23,0.3913,1,"{3238, 3853, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
718,1,341,0,0.9659825,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2,x]","\frac{b^2 \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^2-5 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{b \left(4 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-5 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{b \left(4 a^2-5 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^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}","\frac{b^2 \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^2-5 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{b \left(4 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-5 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{b \left(4 a^2-5 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^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"(b*(4*a^2 - 5*b^2)*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 - 5*b^2)*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^2*(7*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) - (b*(4*a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",12,10,23,0.4348,1,"{3238, 3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
719,1,277,0,0.7111995,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2,x]","\frac{b^2 \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^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{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^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}","\frac{b^2 \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^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{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^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{b \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"-(((2*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + (b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*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 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3238, 3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
720,1,217,0,0.4354403,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^2,x]","\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\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 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}","\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\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 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}",1,"-((b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d)) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + ((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3238, 3845, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
721,1,208,0,0.4197669,"\int \frac{1}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{a \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{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}","-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{a \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{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}",1,"(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + (a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b*(a + b)^2*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3238, 3844, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
722,1,223,0,0.4118309,"\int \frac{1}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{a \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^2-2 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)}{b^2 d \left(a^2-b^2\right)}-\frac{a \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{a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}","\frac{a \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^2-2 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)}{b^2 d \left(a^2-b^2\right)}-\frac{a \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{a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"-((a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3238, 3843, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
723,1,245,0,0.4735083,"\int \frac{1}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{a \left(3 a^2-4 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)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-2 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)}{b^2 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{a \left(3 a^2-4 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)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-2 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)}{b^2 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((3*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*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^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3238, 3847, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
724,1,455,0,1.4514783,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]","\frac{b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(-86 a^2 b^2+63 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}","\frac{b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-65 a^2 b^2+24 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(-86 a^2 b^2+63 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"(b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",13,11,23,0.4783,1,"{3238, 3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
725,1,388,0,1.0323325,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]","\frac{b^2 \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^2 \left(11 a^2-5 b^2\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 b^2+8 a^4+15 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(11 a^2-5 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)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 b^2+8 a^4+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-38 a^2 b^2+35 a^4+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}","\frac{b^2 \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^2 \left(11 a^2-5 b^2\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 b^2+8 a^4+15 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(11 a^2-5 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)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-29 a^2 b^2+8 a^4+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(-38 a^2 b^2+35 a^4+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"-((8*a^4 - 29*a^2*b^2 + 15*b^4)*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) + (b*(11*a^2 - 5*b^2)*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) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*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 - 29*a^2*b^2 + 15*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + (b^2*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^2*(11*a^2 - 5*b^2)*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",12,11,23,0.4783,1,"{3238, 3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
726,1,321,0,0.7702717,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^3,x]","\frac{b^2 \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{3 b^2 \left(3 a^2-b^2\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-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)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-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)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{3 \left(-2 a^2 b^2+5 a^4+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}","\frac{b^2 \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{3 b^2 \left(3 a^2-b^2\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-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)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-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)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{3 \left(-2 a^2 b^2+5 a^4+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}",1,"(-3*b*(3*a^2 - b^2)*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 - b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*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*(a + b)^3*d) + (b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (3*b^2*(3*a^2 - b^2)*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",11,10,23,0.4348,1,"{3238, 3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
727,1,317,0,0.7515855,"\int \frac{1}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{b^2 \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{b \left(7 a^2-b^2\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{3 \left(a^2+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)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+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)}{4 a d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 b^2+3 a^4-b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}","\frac{b^2 \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{b \left(7 a^2-b^2\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{3 \left(a^2+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)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+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)}{4 a d \left(a^2-b^2\right)^2}-\frac{\left(10 a^2 b^2+3 a^4-b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}",1,"((5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*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*(a + b)^3*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) - (b*(7*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3238, 3845, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
728,1,302,0,0.6964768,"\int \frac{1}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{3 \left(a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{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{a \left(a^2-7 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)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(a^2+5 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)}{4 b d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 b^2+a^4-3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}","\frac{3 \left(a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{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{a \left(a^2-7 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)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(a^2+5 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)}{4 b d \left(a^2-b^2\right)^2}-\frac{\left(-10 a^2 b^2+a^4-3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}",1,"-((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d) + (a*(a^2 - 7*b^2)*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) - ((a^4 - 10*a^2*b^2 - 3*b^4)*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^2*(a + b)^3*d) - (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 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3238, 3844, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
729,1,319,0,0.6850239,"\int \frac{1}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{a \left(a^2-7 b^2\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 \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(-5 a^2 b^2+3 a^4+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{3 a \left(a^2-3 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)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{3 a \left(-2 a^2 b^2+a^4+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}","\frac{a \left(a^2-7 b^2\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 \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(-5 a^2 b^2+3 a^4+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{3 a \left(a^2-3 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)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{3 a \left(-2 a^2 b^2+a^4+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"(-3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*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) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (a*(a^2 - 7*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3238, 3843, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
730,1,369,0,0.7527613,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-2 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+2 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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-2 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+2 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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2 - 2*b^2)*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 + 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)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,25,0.2400,1,"{4222, 2796, 3055, 2998, 2816, 2994}"
731,1,311,0,0.5019098,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2),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)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \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} \sqrt{\cos (c+d x)} \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 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \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} \sqrt{\cos (c+d x)} \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)}}",1,"(2*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*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)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{4222, 2796, 2998, 2816, 2994}"
732,1,269,0,0.3598251,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2),x]","\frac{2 (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 (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}} F\left(\sin ^{-1}\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-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 (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}} F\left(\sin ^{-1}\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]*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*(a - b)*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]])","A",4,4,25,0.1600,1,"{4222, 2795, 2816, 2994}"
733,1,155,0,0.1434332,"\int \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]],x]","-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)} \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}","-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)} \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(-2*Sqrt[Cos[c + d*x]]*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + d*x]))/(a + b*Cos[c + d*x])]*(a + b*Cos[c + d*x])*Csc[c + d*x]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]]], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(Sqrt[a + b]*d)","A",2,2,25,0.08000,1,"{4222, 2811}"
734,1,431,0,0.6498341,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{\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)}{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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{a \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{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{\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)}{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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"-(((a - b)*Sqrt[a + b]*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]*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]]) - (a*Sqrt[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]]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,25,0.3200,1,"{4222, 2821, 3054, 2809, 12, 2801, 2816, 2994}"
735,1,498,0,0.991045,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b} \left(a^2-4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \left(a^2-4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a + 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)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a^2 - 4*b^2)*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]]) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)","A",8,8,25,0.3200,1,"{4222, 2821, 3061, 3053, 2809, 2998, 2816, 2994}"
736,1,427,0,1.0499851,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2),x]","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{4 b (a-b) \sqrt{a+b} \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{4 b (a-b) \sqrt{a+b} \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}",1,"(4*(a - b)*b*Sqrt[a + b]*(41*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(25*a^2 - 57*a*b - 6*b^2)*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 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (16*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,25,0.2400,1,"{4222, 2799, 3055, 2998, 2816, 2994}"
737,1,365,0,0.759106,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 (a-b) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}","\frac{2 (a-b) \sqrt{a+b} \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 (a-b) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}",1,"(2*(a - b)*Sqrt[a + b]*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*(3*a - b)*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)])/(5*a*d*Sqrt[Sec[c + d*x]]) + (4*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,25,0.2400,1,"{4222, 2799, 3055, 2998, 2816, 2994}"
738,1,317,0,0.5260502,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2),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-3 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}} F\left(\sin ^{-1}\left(\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{8 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a 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-3 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}} F\left(\sin ^{-1}\left(\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{8 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}",1,"(8*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*(a - 3*b)*(a - b)*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)])/(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",5,5,25,0.2000,1,"{4222, 2799, 2998, 2816, 2994}"
739,1,397,0,0.555988,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2),x]","-\frac{2 (a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}+\frac{2 (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)}{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 (a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}+\frac{2 (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)}{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 - 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)])/(d*Sqrt[Sec[c + d*x]]) - (2*(a - 2*b)*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)])/(d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[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]])","A",6,6,25,0.2400,1,"{4222, 2798, 2809, 2998, 2816, 2994}"
740,1,435,0,0.7154591,"\int (a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]],x]","\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\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{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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\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{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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"-(((a - b)*b*Sqrt[a + b]*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]]) - (3*a*Sqrt[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]]) + (b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,7,25,0.2800,1,"{4222, 2821, 3053, 2809, 2998, 2816, 2994}"
741,1,493,0,1.270196,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{2 d}+\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{2 d}+\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}",1,"(-5*(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)])/(4*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a + 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)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*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]]) + (3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + ((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",9,9,25,0.3600,1,"{4222, 2821, 3047, 3061, 3053, 2809, 2998, 2816, 2994}"
742,1,568,0,1.3556632,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{\left(3 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 d \sqrt{\sec (c+d x)}}","\frac{\left(3 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(3*a^2 + 16*b^2)*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]*(a + 2*b)*(3*a + 8*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]]) + (a*Sqrt[a + b]*(a^2 - 12*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + ((a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((3*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)","A",9,9,25,0.3600,1,"{4222, 2821, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
743,1,494,0,1.5175968,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \left(-114 a^2 b+147 a^3+165 a b^2+10 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 b^2+147 a^4-10 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^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}+\frac{38 a b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \left(-114 a^2 b+147 a^3+165 a b^2+10 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 b^2+147 a^4-10 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^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}+\frac{38 a b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}",1,"(2*(a - b)*Sqrt[a + b]*(147*a^4 + 279*a^2*b^2 - 10*b^4)*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]*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (38*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",8,6,25,0.2400,1,"{4222, 2792, 3055, 2998, 2816, 2994}"
744,1,427,0,1.1673059,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2),x]","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 a b+3 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)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{6 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 a b+3 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)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{6 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}",1,"(2*(a - b)*b*Sqrt[a + b]*(29*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(5*a^2 - 24*a*b + 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (6*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",7,6,25,0.2400,1,"{4222, 2792, 3055, 2998, 2816, 2994}"
745,1,378,0,0.8651774,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-8 a b+15 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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{22 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} \left(9 a^2-8 a b+15 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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{22 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(a - b)*Sqrt[a + b]*(9*a^2 + 23*b^2)*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*(a - b)*Sqrt[a + b]*(9*a^2 - 8*a*b + 15*b^2)*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]]) + (22*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",6,6,25,0.2400,1,"{4222, 2792, 3055, 2998, 2816, 2994}"
746,1,452,0,0.8236246,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b^2 \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{14 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b^2 \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{14 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d \sqrt{\sec (c+d x)}}",1,"(14*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(a^2 - 7*a*b + 9*b^2)*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]]) - (2*b^2*Sqrt[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*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,7,25,0.2800,1,"{4222, 2792, 3053, 2809, 2998, 2816, 2994}"
747,1,505,0,1.1070387,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2),x]","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{5 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)}{d \sqrt{\sec (c+d x)}}","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{5 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)}{d \sqrt{\sec (c+d x)}}",1,"((a - b)*Sqrt[a + b]*(2*a^2 - b^2)*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^2 - 6*a*b - b^2)*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]]) - (5*a*b*Sqrt[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*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",8,8,25,0.3200,1,"{4222, 2792, 3061, 3053, 2809, 2998, 2816, 2994}"
748,1,503,0,1.1009061,"\int (a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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)}{4 d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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)}{4 d \sqrt{\sec (c+d x)}}",1,"(-9*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^2 + 9*a*b + 2*b^2)*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]*(15*a^2 + 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",8,8,25,0.3200,1,"{4222, 2793, 3061, 3053, 2809, 2998, 2816, 2994}"
749,1,566,0,1.4509512,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}",1,"-((a - b)*Sqrt[a + b]*(33*a^2 + 16*b^2)*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]*(33*a^2 + 26*a*b + 16*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (5*a*Sqrt[a + b]*(a^2 + 4*b^2)*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*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (13*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Sec[c + d*x]]) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d)","A",9,9,25,0.3600,1,"{4222, 2793, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
750,1,638,0,1.8826159,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{\left(59 a^2+36 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d \sqrt{\sec (c+d x)}}+\frac{a \left(15 a^2+284 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sqrt{a+b} \left(118 a^2 b+15 a^3+284 a b^2+72 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)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2+284 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)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-120 a^2 b^2+5 a^4-48 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{17 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\left(59 a^2+36 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d \sqrt{\sec (c+d x)}}+\frac{a \left(15 a^2+284 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sqrt{a+b} \left(118 a^2 b+15 a^3+284 a b^2+72 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)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2+284 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)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-120 a^2 b^2+5 a^4-48 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{17 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sec ^{\frac{3}{2}}(c+d x)}",1,"-((a - b)*Sqrt[a + b]*(15*a^2 + 284*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3 + 118*a^2*b + 284*a*b^2 + 72*b^3)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4 - 120*a^2*b^2 - 48*b^4)*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^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (17*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + ((59*a^2 + 36*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sqrt[Sec[c + d*x]]) + (a*(15*a^2 + 284*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)","A",10,9,25,0.3600,1,"{4222, 2793, 3049, 3061, 3053, 2809, 2998, 2816, 2994}"
751,1,314,0,0.4848214,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b} (a+2 b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \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+2 b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(-4*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(a + 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)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)","A",5,5,25,0.2000,1,"{4222, 2802, 2998, 2816, 2994}"
752,1,264,0,0.3091207,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 (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} \sqrt{\cos (c+d x)} \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-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} \sqrt{\cos (c+d x)} \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]*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]*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",4,4,25,0.1600,1,"{4222, 2801, 2816, 2994}"
753,1,129,0,0.1308484,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \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 \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]*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",2,2,25,0.08000,1,"{4222, 2816}"
754,1,136,0,0.1320363,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{2 \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 \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]*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",2,2,25,0.08000,1,"{4222, 2809}"
755,1,474,0,0.7962908,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\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{(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 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\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{(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]*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]*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]*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]]) + Sin[c + d*x]/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",9,9,25,0.3600,1,"{4222, 2820, 2809, 3003, 2993, 12, 2801, 2816, 2994}"
756,1,505,0,0.9543004,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)),x]","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{(3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\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+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{(3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(3*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - ((3*a - 2*b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*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]]) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)","A",8,8,25,0.3200,1,"{4222, 2793, 3061, 3053, 2809, 2998, 2816, 2994}"
757,1,397,0,0.8232701,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 (a+2 b) (a+4 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 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 (a+2 b) (a+4 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*b*(5*a^2 - 8*b^2)*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)*(a + 4*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^2*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 - 4*b^2)*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",6,6,25,0.2400,1,"{4222, 2802, 3055, 2998, 2816, 2994}"
758,1,325,0,0.5489951,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 b^2 \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^2-2 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+2 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^2 \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^2-2 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+2 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 - 2*b^2)*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*(a + 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)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2802, 2998, 2816, 2994}"
759,1,307,0,0.4790456,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 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 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 \sqrt{\cos (c+d x)} \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 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 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 \sqrt{\cos (c+d x)} \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*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*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*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2800, 2998, 2816, 2994}"
760,1,306,0,0.418758,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{2 a \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 \sqrt{\cos (c+d x)} \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 \sqrt{\cos (c+d x)} \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{a+b} \sqrt{\sec (c+d x)}}","\frac{2 a \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 \sqrt{\cos (c+d x)} \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 \sqrt{\cos (c+d x)} \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{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*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*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*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*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4222, 2794, 2795, 2816, 2994}"
761,1,447,0,0.5952907,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 a^2 \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 \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 \sqrt{\cos (c+d x)} \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{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \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)}{b d \sqrt{a+b} \sqrt{\sec (c+d x)}}","-\frac{2 a^2 \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 \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 \sqrt{\cos (c+d x)} \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{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \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)}{b d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(2*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)])/(b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*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*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[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]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",7,7,25,0.2800,1,"{4222, 2797, 2809, 2794, 2795, 2816, 2994}"
762,1,525,0,1.0909398,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)),x]","-\frac{2 a^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{(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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{(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{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\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,"-(((3*a^2 - b^2)*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]])) + ((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)])/(b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (3*a*Sqrt[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^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",8,8,25,0.3200,1,"{4222, 2792, 3061, 3053, 2809, 2998, 2816, 2994}"
763,1,513,0,1.3017338,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{4 b^2 \left(5 a^2-3 b^2\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^2 \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(-13 a^2 b^2+a^4+8 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 \left(16 a^2 b^2+9 a^3 b+a^4-12 a b^3-16 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{4 b^2 \left(5 a^2-3 b^2\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^2 \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(-13 a^2 b^2+a^4+8 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 \left(16 a^2 b^2+9 a^3 b+a^4-12 a b^3-16 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*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*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*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 - 13*a^2*b^2 + 8*b^4)*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",7,6,25,0.2400,1,"{4222, 2802, 3055, 2998, 2816, 2994}"
764,1,438,0,0.9251466,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{8 b^2 \left(2 a^2-b^2\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^2 \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(9 a^2 b+3 a^3-6 a b^2-8 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 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{8 b^2 \left(2 a^2-b^2\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^2 \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(9 a^2 b+3 a^3-6 a b^2-8 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 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 - 15*a^2*b^2 + 8*b^4)*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*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*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",6,6,25,0.2400,1,"{4222, 2802, 3055, 2998, 2816, 2994}"
765,1,421,0,0.8440573,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2-3 a b-2 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{4 b \left(3 a^2-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^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2-3 a b-2 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{4 b \left(3 a^2-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^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(4*b*(3*a^2 - b^2)*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*(3*a^2 - 3*a*b - 2*b^2)*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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (4*b*(3*a^2 - b^2)*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",6,6,25,0.2400,1,"{4222, 2802, 2993, 2998, 2816, 2994}"
766,1,399,0,0.7575625,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{2 \left(3 a^2+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 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+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 (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)}{3 a d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","\frac{2 \left(3 a^2+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 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+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 (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)}{3 a d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-2*(3*a^2 + b^2)*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*(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*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*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 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{4222, 2796, 2993, 2998, 2816, 2994}"
767,1,382,0,0.7288969,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","-\frac{8 a b \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 \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 (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{8 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}","-\frac{8 a b \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 \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 (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{8 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(8*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)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (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)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (8*a*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",6,6,25,0.2400,1,"{4222, 2799, 2993, 2998, 2816, 2994}"
768,1,557,0,1.2128394,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","-\frac{2 a^2 \left(3 a^2-7 b^2\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^2 \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(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 \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^2 \left(3 a^2-7 b^2\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^2 \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(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 \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*(3*a^2 - 7*b^2)*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 - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(3*a^2 + a*b - 6*b^2)*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^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[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^2*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^2*(3*a^2 - 7*b^2)*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",8,8,25,0.3200,1,"{4222, 2792, 3051, 2809, 2993, 2998, 2816, 2994}"
769,1,330,0,0.6742378,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^4 \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^4,x]","-\frac{\left(6 a^2 b^2 \left(m^2+5 m+4\right)+a^4 \left(m^2+6 m+8\right)+b^4 \left(m^2+4 m+3\right)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{4 a b \left(a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \left(a^2 (5 m+22)+b^2 (m+3)\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b^3 (m+5) \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}","-\frac{\left(6 a^2 b^2 \left(m^2+5 m+4\right)+a^4 \left(m^2+6 m+8\right)+b^4 \left(m^2+4 m+3\right)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{4 a b \left(a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \left(a^2 (5 m+22)+b^2 (m+3)\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b^3 (m+5) \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}",1,"(b^2*(b^2*(3 + m) + a^2*(22 + 5*m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b^3*(5 + m)*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b^2*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((b^4*(3 + 4*m + m^2) + 6*a^2*b^2*(4 + 5*m + m^2) + a^4*(8 + 6*m + m^2))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (4*a*b*(b^2*(2 + m) + a^2*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",6,5,21,0.2381,1,"{2793, 3033, 3023, 2748, 2643}"
770,1,250,0,0.3166594,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^3 \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^3,x]","-\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)}","-\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)}",1,"(a*b^2*(7 + 2*m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (b^2*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)) - (a*(3*b^2*(1 + m) + a^2*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (b*(b^2*(2 + m) + 3*a^2*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","A",5,4,21,0.1905,1,"{2793, 3023, 2748, 2643}"
771,1,179,0,0.1275356,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2,x]","-\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}","-\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}",1,"(b^2*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) - ((b^2*(1 + m) + a^2*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",4,3,21,0.1429,1,"{2789, 2643, 3014}"
772,1,131,0,0.0640834,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x]),x]","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"-((a*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (b*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])","A",3,2,19,0.1053,1,"{2748, 2643}"
773,1,190,0,0.2326857,"\int \frac{\cos ^m(c+d x)}{a+b \cos (c+d x)} \, dx","Int[Cos[c + d*x]^m/(a + b*Cos[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}","\frac{a \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"(a*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/((a^2 - b^2)*d) - (b*AppellF1[1/2, -m/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(m/2))","A",5,3,21,0.1429,1,"{2823, 3189, 429}"
774,1,294,0,0.3535101,"\int \frac{\cos ^m(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Int[Cos[c + d*x]^m/(a + b*Cos[c + d*x])^2,x]","\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) \cos ^2(c+d x)^{\frac{1}{2} (-m-1)} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}+\frac{a^2 \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}","\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) \cos ^2(c+d x)^{\frac{1}{2} (-m-1)} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}+\frac{a^2 \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}",1,"(b^2*AppellF1[1/2, (-1 - m)/2, 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(1 + m)*(Cos[c + d*x]^2)^((-1 - m)/2)*Sin[c + d*x])/((a^2 - b^2)^2*d) + (a^2*AppellF1[1/2, (1 - m)/2, 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/((a^2 - b^2)^2*d) - (2*a*b*AppellF1[1/2, -m/2, 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((a^2 - b^2)^2*d*(Cos[c + d*x]^2)^(m/2))","A",8,3,21,0.1429,1,"{2824, 3189, 429}"
775,1,282,0,0.4240417,"\int (a+b \cos (c+d x))^3 \sec ^m(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^m,x]","-\frac{b \left(3 a^2 (3-m)+b^2 (2-m)\right) \sin (c+d x) \sec ^{m-4}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(c+d x)\right)}{d (2-m) (4-m) \sqrt{\sin ^2(c+d x)}}-\frac{a \left(a^2 (2-m)+3 b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-2}(c+d x) (a \sec (c+d x)+b)}{d (1-m)}-\frac{a^2 b (1-2 m) \sin (c+d x) \sec ^{m-2}(c+d x)}{d (1-m) (2-m)}","-\frac{b \left(3 a^2 (3-m)+b^2 (2-m)\right) \sin (c+d x) \sec ^{m-4}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(c+d x)\right)}{d (2-m) (4-m) \sqrt{\sin ^2(c+d x)}}-\frac{a \left(a^2 (2-m)+3 b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-2}(c+d x) (a \sec (c+d x)+b)}{d (1-m)}-\frac{a^2 b (1-2 m) \sin (c+d x) \sec ^{m-2}(c+d x)}{d (1-m) (2-m)}",1,"-((a^2*b*(1 - 2*m)*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(1 - m)*(2 - m))) - (a^2*Sec[c + d*x]^(-2 + m)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(d*(1 - m)) - (b*(b^2*(2 - m) + 3*a^2*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-4 + m)*Sin[c + d*x])/(d*(2 - m)*(4 - m)*Sqrt[Sin[c + d*x]^2]) - (a*(3*b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(1 - m)*(3 - m)*Sqrt[Sin[c + d*x]^2])","A",8,6,21,0.2857,1,"{3238, 3842, 4047, 3772, 2643, 4046}"
776,1,200,0,0.1885667,"\int (a+b \cos (c+d x))^2 \sec ^m(c+d x) \, dx","Int[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^m,x]","-\frac{\left(a^2 (2-m)+b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-1}(c+d x)}{d (1-m)}-\frac{2 a b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}","-\frac{\left(a^2 (2-m)+b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-1}(c+d x)}{d (1-m)}-\frac{2 a b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}",1,"-((a^2*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m))) - ((b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(1 - m)*(3 - m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - m)*Sqrt[Sin[c + d*x]^2])","A",7,5,21,0.2381,1,"{3238, 3788, 3772, 2643, 4046}"
777,1,143,0,0.1121459,"\int (a+b \cos (c+d x)) \sec ^m(c+d x) \, dx","Int[(a + b*Cos[c + d*x])*Sec[c + d*x]^m,x]","-\frac{a \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d (1-m) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}","-\frac{a \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d (1-m) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}",1,"-((b*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - m)*Sqrt[Sin[c + d*x]^2])) - (a*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m)*Sqrt[Sin[c + d*x]^2])","A",6,4,19,0.2105,1,"{3238, 3787, 3772, 2643}"
778,1,26,0,0.0776947,"\int \frac{\sqrt{1-\cos (x)}}{\sqrt{a-\cos (x)}} \, dx","Int[Sqrt[1 - Cos[x]]/Sqrt[a - Cos[x]],x]","-2 \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)","-2 \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)",1,"-2*ArcTan[Sin[x]/(Sqrt[1 - Cos[x]]*Sqrt[a - Cos[x]])]","A",2,2,21,0.09524,1,"{2775, 204}"
779,1,65,0,0.0999325,"\int \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \, dx","Int[Sqrt[(1 - Cos[x])/(a - Cos[x])],x]","-\frac{2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)} \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)}{\sqrt{1-\cos (x)}}","-\frac{2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)} \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)}{\sqrt{1-\cos (x)}}",1,"(-2*ArcTan[Sin[x]/(Sqrt[1 - Cos[x]]*Sqrt[a - Cos[x]])]*Sqrt[(1 - Cos[x])/(a - Cos[x])]*Sqrt[a - Cos[x]])/Sqrt[1 - Cos[x]]","A",3,3,19,0.1579,1,"{4400, 2775, 204}"
780,1,37,0,0.0201143,"\int (a+a \cos (c+d x)) \left(-\frac{B}{2}+B \cos (c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])*(-B/2 + B*Cos[c + d*x]),x]","\frac{a B \sin (c+d x)}{2 d}+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a B \sin (c+d x)}{2 d}+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*B*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,25,0.04000,1,"{2734}"
781,1,26,0,0.0313192,"\int (a+a \cos (c+d x))^4 \left(-\frac{4 B}{5}+B \cos (c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^4*((-4*B)/5 + B*Cos[c + d*x]),x]","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)","A",1,1,27,0.03704,1,"{2749}"
782,1,28,0,0.0447429,"\int (a+a \cos (c+d x))^n \left(-\frac{B n}{1+n}+B \cos (c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^n*(-((B*n)/(1 + n)) + B*Cos[c + d*x]),x]","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^n}{d (n+1)}","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^n}{d (n+1)}",1,"(B*(a + a*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + n))","A",1,1,31,0.03226,1,"{2749}"
783,1,26,0,0.0328949,"\int \frac{-\frac{3 B}{2}+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Int[((-3*B)/2 + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{B \sin (c+d x)}{2 d (a \cos (c+d x)+a)^3}","-\frac{B \sin (c+d x)}{2 d (a \cos (c+d x)+a)^3}",1,"-(B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^3)","A",1,1,27,0.03704,1,"{2749}"
784,1,28,0,0.0424421,"\int (a+a \cos (c+d x))^{3/2} \left(-\frac{3 B}{5}+B \cos (c+d x)\right) \, dx","Int[(a + a*Cos[c + d*x])^(3/2)*((-3*B)/5 + B*Cos[c + d*x]),x]","\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}","\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",1,1,29,0.03448,1,"{2749}"
785,1,26,0,0.0155015,"\int \frac{B+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Int[(B + B*Cos[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}","\frac{2 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])","A",2,2,25,0.08000,1,"{21, 2646}"
786,1,28,0,0.0423808,"\int \frac{-\frac{5 B}{3}+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Int[((-5*B)/3 + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{2 B \sin (c+d x)}{3 d (a \cos (c+d x)+a)^{5/2}}","-\frac{2 B \sin (c+d x)}{3 d (a \cos (c+d x)+a)^{5/2}}",1,"(-2*B*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^(5/2))","A",1,1,29,0.03448,1,"{2749}"
787,1,104,0,0.0819791,"\int (a+a \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","\frac{2 \sqrt[6]{2} (5 A+2 B) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{5 d (\cos (c+d x)+1)^{7/6}}+\frac{3 B \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 d}","\frac{2 \sqrt[6]{2} (5 A+2 B) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{5 d (\cos (c+d x)+1)^{7/6}}+\frac{3 B \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 d}",1,"(3*B*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d) + (2*2^(1/6)*(5*A + 2*B)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(5*d*(1 + Cos[c + d*x])^(7/6))","A",3,3,25,0.1200,1,"{2751, 2652, 2651}"
788,1,102,0,0.0789982,"\int \sqrt[3]{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[(a + a*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","\frac{(4 A+B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2 \sqrt[6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 B \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{4 d}","\frac{(4 A+B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2 \sqrt[6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 B \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{4 d}",1,"(3*B*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d) + ((4*A + B)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(2*2^(1/6)*d*(1 + Cos[c + d*x])^(5/6))","A",3,3,25,0.1200,1,"{2751, 2652, 2651}"
789,1,101,0,0.0771416,"\int \frac{A+B \cos (c+d x)}{\sqrt[3]{a+a \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/3),x]","\frac{(2 A-B) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2^{5/6} d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 B \sin (c+d x)}{2 d \sqrt[3]{a \cos (c+d x)+a}}","\frac{(2 A-B) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2^{5/6} d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 B \sin (c+d x)}{2 d \sqrt[3]{a \cos (c+d x)+a}}",1,"(3*B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(1/3)) + ((2*A - B)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(2^(5/6)*d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))","A",3,3,25,0.1200,1,"{2751, 2652, 2651}"
790,1,105,0,0.0894841,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(2/3),x]","\frac{3 (A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}-\frac{2^{5/6} (A-2 B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{a d (\cos (c+d x)+1)^{5/6}}","\frac{3 (A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}-\frac{2^{5/6} (A-2 B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{a d (\cos (c+d x)+1)^{5/6}}",1,"(3*(A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(2/3)) - (2^(5/6)*(A - 2*B)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x])^(5/6))","A",3,3,25,0.1200,1,"{2750, 2652, 2651}"
791,1,63,0,0.0931029,"\int \frac{\frac{b B}{a}+B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Int[((b*B)/a + B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{B x}{b}-\frac{2 B \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}","\frac{B x}{b}-\frac{2 B \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}",1,"(B*x)/b - (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*b*d)","A",3,3,28,0.1071,1,"{2735, 2659, 205}"
792,1,22,0,0.0311741,"\int \frac{a+b \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx","Int[(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2,x]","\frac{\sin (c+d x)}{d (a \cos (c+d x)+b)}","\frac{\sin (c+d x)}{d (a \cos (c+d x)+b)}",1,"Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))","A",2,2,23,0.08696,1,"{2754, 8}"
793,1,47,0,0.0670049,"\int \frac{3+\cos (c+d x)}{2-\cos (c+d x)} \, dx","Int[(3 + Cos[c + d*x])/(2 - Cos[c + d*x]),x]","\frac{10 \tan ^{-1}\left(\frac{\sin (c+d x)}{-\cos (c+d x)+\sqrt{3}+2}\right)}{\sqrt{3} d}+\frac{5 x}{\sqrt{3}}-x","\frac{10 \tan ^{-1}\left(\frac{\sin (c+d x)}{-\cos (c+d x)+\sqrt{3}+2}\right)}{\sqrt{3} d}+\frac{5 x}{\sqrt{3}}-x",1,"-x + (5*x)/Sqrt[3] + (10*ArcTan[Sin[c + d*x]/(2 + Sqrt[3] - Cos[c + d*x])])/(Sqrt[3]*d)","A",2,2,21,0.09524,1,"{2735, 2657}"
794,1,58,0,0.0447661,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Int[(a*B + b*B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]],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 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)])","A",3,3,28,0.1071,1,"{21, 2655, 2653}"
795,1,229,0,0.2228513,"\int (a+b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{2} (A b-a B) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}",1,"(Sqrt[2]*(a + b)*B*AppellF1[1/2, 1/2, -5/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))","A",7,4,25,0.1600,1,"{2756, 2665, 139, 138}"
796,1,229,0,0.1890729,"\int \sqrt[3]{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[(a + b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(Sqrt[2]*(a + b)*B*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))","A",7,4,25,0.1600,1,"{2756, 2665, 139, 138}"
797,1,226,0,0.1830025,"\int \frac{A+B \cos (c+d x)}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3),x]","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}+\frac{\sqrt{2} B \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}+\frac{\sqrt{2} B \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}",1,"(Sqrt[2]*B*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))","A",7,4,25,0.1600,1,"{2756, 2665, 139, 138}"
798,1,226,0,0.1850213,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{2/3}} \, dx","Int[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(2/3),x]","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}+\frac{\sqrt{2} B \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}+\frac{\sqrt{2} B \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(Sqrt[2]*B*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Cos[c + d*x])/2, (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))","A",7,4,25,0.1600,1,"{2756, 2665, 139, 138}"
799,1,168,0,0.1423519,"\int \cos ^2(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d)","A",9,7,31,0.2258,1,"{16, 2748, 2635, 2640, 2639, 2642, 2641}"
800,1,139,0,0.1167908,"\int \cos (c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)","A",8,7,29,0.2414,1,"{16, 2748, 2635, 2642, 2641, 2640, 2639}"
801,1,108,0,0.0835939,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,23,0.2609,1,"{2748, 2640, 2639, 2635, 2642, 2641}"
802,1,80,0,0.0867523,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])","A",6,6,29,0.2069,1,"{16, 2748, 2642, 2641, 2640, 2639}"
803,1,105,0,0.1172933,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 A b \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(-2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",7,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
804,1,136,0,0.1313993,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 A b^2 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(-2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",8,7,31,0.2258,1,"{16, 2748, 2636, 2642, 2641, 2640, 2639}"
805,1,169,0,0.1625295,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])","A",9,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
806,1,169,0,0.1389168,"\int \cos (c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}",1,"(6*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*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*b*d)","A",9,7,29,0.2414,1,"{16, 2748, 2635, 2640, 2639, 2642, 2641}"
807,1,140,0,0.1040176,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(6*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sqrt[b*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",7,6,23,0.2609,1,"{2748, 2635, 2642, 2641, 2640, 2639}"
808,1,112,0,0.1036827,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}","\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(2*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,7,29,0.2414,1,"{16, 2748, 2640, 2639, 2635, 2642, 2641}"
809,1,83,0,0.1006453,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])","A",6,6,31,0.1935,1,"{16, 2748, 2642, 2641, 2640, 2639}"
810,1,110,0,0.1204012,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(-2*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",7,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
811,1,141,0,0.1411574,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 A b^3 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(-2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",8,7,31,0.2258,1,"{16, 2748, 2636, 2642, 2641, 2640, 2639}"
812,1,174,0,0.157485,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-6*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^3*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b^2*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])","A",9,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
813,1,171,0,0.1197544,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}","\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}",1,"(6*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*b*(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",8,6,23,0.2609,1,"{2748, 2635, 2640, 2639, 2642, 2641}"
814,1,145,0,0.1234822,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{6 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}","\frac{2 A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{6 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(6*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",8,7,29,0.2414,1,"{16, 2748, 2635, 2642, 2641, 2640, 2639}"
815,1,116,0,0.1137402,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,7,31,0.2258,1,"{16, 2748, 2640, 2639, 2635, 2642, 2641}"
816,1,85,0,0.0986461,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])","A",6,6,31,0.1935,1,"{16, 2748, 2642, 2641, 2640, 2639}"
817,1,112,0,0.1203016,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 A b^3 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(-2*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",7,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
818,1,143,0,0.1382124,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{2 A b^4 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(-2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^3*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",8,7,31,0.2258,1,"{16, 2748, 2636, 2642, 2641, 2640, 2639}"
819,1,176,0,0.1620023,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Int[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{2 A b^5 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^3 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^4 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^5 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^3 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^4 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-6*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^5*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^4*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b^3*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])","A",9,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
820,1,173,0,0.1349035,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d)","A",9,7,31,0.2258,1,"{16, 2748, 2635, 2640, 2639, 2642, 2641}"
821,1,144,0,0.1133659,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}",1,"(6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d)","A",8,7,31,0.2258,1,"{16, 2748, 2635, 2642, 2641, 2640, 2639}"
822,1,113,0,0.0925369,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)","A",7,7,29,0.2414,1,"{16, 2748, 2640, 2639, 2635, 2642, 2641}"
823,1,82,0,0.0677927,"\int \frac{A+B \cos (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"(2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])","A",5,5,23,0.2174,1,"{2748, 2642, 2641, 2640, 2639}"
824,1,106,0,0.1034445,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(-2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",7,7,29,0.2414,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
825,1,135,0,0.1301978,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}","\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"(-2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",8,7,31,0.2258,1,"{16, 2748, 2636, 2642, 2641, 2640, 2639}"
826,1,168,0,0.1535724,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])","A",9,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
827,1,176,0,0.1351437,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}",1,"(6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d)","A",9,7,31,0.2258,1,"{16, 2748, 2635, 2640, 2639, 2642, 2641}"
828,1,147,0,0.1128997,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}",1,"(6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d)","A",8,7,31,0.2258,1,"{16, 2748, 2635, 2642, 2641, 2640, 2639}"
829,1,116,0,0.0940435,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)","A",7,7,31,0.2258,1,"{16, 2748, 2640, 2639, 2635, 2642, 2641}"
830,1,85,0,0.0772422,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"(2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b*d*Sqrt[b*Cos[c + d*x]])","A",6,6,29,0.2069,1,"{16, 2748, 2642, 2641, 2640, 2639}"
831,1,112,0,0.0885878,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}",1,"(-2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])","A",6,6,23,0.2609,1,"{2748, 2636, 2640, 2639, 2642, 2641}"
832,1,140,0,0.1274983,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}",1,"(-2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])","A",8,7,29,0.2414,1,"{16, 2748, 2636, 2642, 2641, 2640, 2639}"
833,1,171,0,0.1600714,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{6 A \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{6 A \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(-6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])","A",9,7,31,0.2258,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
834,1,176,0,0.1337192,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^5*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}",1,"(6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)","A",9,7,31,0.2258,1,"{16, 2748, 2635, 2640, 2639, 2642, 2641}"
835,1,147,0,0.1143526,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}",1,"(6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d)","A",8,7,31,0.2258,1,"{16, 2748, 2635, 2642, 2641, 2640, 2639}"
836,1,116,0,0.0910747,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d)","A",7,7,31,0.2258,1,"{16, 2748, 2640, 2639, 2635, 2642, 2641}"
837,1,85,0,0.0746879,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}",1,"(2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",6,6,31,0.1935,1,"{16, 2748, 2642, 2641, 2640, 2639}"
838,1,112,0,0.0965816,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(-2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",7,7,29,0.2414,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
839,1,143,0,0.1017596,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}",1,"(-2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",7,6,23,0.2609,1,"{2748, 2636, 2642, 2641, 2640, 2639}"
840,1,173,0,0.1437792,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{6 A \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}","\frac{6 A \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}",1,"(-6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])","A",9,7,29,0.2414,1,"{16, 2748, 2636, 2640, 2639, 2642, 2641}"
841,1,176,0,0.1191402,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx","Int[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(7/2),x]","\frac{6 A \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d \sqrt{b \cos (c+d x)}}","\frac{6 A \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d \sqrt{b \cos (c+d x)}}",1,"(-6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b^2*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])","A",8,6,23,0.2609,1,"{2748, 2636, 2640, 2639, 2642, 2641}"
842,1,172,0,0.0685106,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","-\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}","-\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"(3*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",7,5,33,0.1515,1,"{17, 2748, 2633, 2635, 8}"
843,1,136,0,0.0550547,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{A x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
844,1,98,0,0.0222809,"\int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",2,2,33,0.06061,1,"{17, 2734}"
845,1,59,0,0.012328,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
846,1,60,0,0.0258115,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}","\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"(B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])","A",3,3,33,0.09091,1,"{17, 2735, 3770}"
847,1,68,0,0.0405146,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",5,5,33,0.1515,1,"{17, 2748, 3767, 8, 3770}"
848,1,107,0,0.0541124,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",6,6,33,0.1818,1,"{17, 2748, 3768, 3770, 3767, 8}"
849,1,145,0,0.0613371,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}","\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))","A",6,5,33,0.1515,1,"{17, 2748, 3767, 3768, 3770}"
850,1,177,0,0.0694481,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","-\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}","-\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"(3*b*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",7,5,33,0.1515,1,"{17, 2748, 2633, 2635, 8}"
851,1,140,0,0.0562753,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{A b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*b*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
852,1,101,0,0.0229203,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(b*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",2,2,33,0.06061,1,"{17, 2734}"
853,1,61,0,0.0129245,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{A b x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A b x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*b*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
854,1,62,0,0.0264272,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}","\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"(b*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])","A",3,3,33,0.09091,1,"{17, 2735, 3770}"
855,1,70,0,0.0405489,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",5,5,33,0.1515,1,"{17, 2748, 3767, 8, 3770}"
856,1,110,0,0.0551959,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",6,6,33,0.1818,1,"{17, 2748, 3768, 3770, 3767, 8}"
857,1,149,0,0.0625064,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}","\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))","A",6,5,33,0.1515,1,"{17, 2748, 3767, 3768, 3770}"
858,1,187,0,0.0719922,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","-\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b^2 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}","-\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b^2 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"(3*b^2*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",7,5,33,0.1515,1,"{17, 2748, 2633, 2635, 8}"
859,1,148,0,0.0563016,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b^2 B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b^2 B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*b^2*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
860,1,107,0,0.0239772,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(b^2*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)","A",2,2,33,0.06061,1,"{17, 2734}"
861,1,65,0,0.0141022,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(A*b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
862,1,66,0,0.0263802,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}","\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"(b^2*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])","A",3,3,33,0.09091,1,"{17, 2735, 3770}"
863,1,74,0,0.0426741,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",5,5,33,0.1515,1,"{17, 2748, 3767, 8, 3770}"
864,1,116,0,0.0569699,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))","A",6,6,33,0.1818,1,"{17, 2748, 3768, 3770, 3767, 8}"
865,1,157,0,0.0635576,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(13/2),x]","\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}","\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))","A",6,5,33,0.1515,1,"{17, 2748, 3767, 3768, 3770}"
866,1,136,0,0.0556648,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{A x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
867,1,98,0,0.0238959,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]])","A",2,2,33,0.06061,1,"{17, 2734}"
868,1,59,0,0.0131365,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
869,1,60,0,0.0264577,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]])","A",3,3,33,0.09091,1,"{18, 2735, 3770}"
870,1,68,0,0.0418939,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",5,5,33,0.1515,1,"{18, 2748, 3767, 8, 3770}"
871,1,107,0,0.0545298,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",6,6,33,0.1818,1,"{18, 2748, 3768, 3770, 3767, 8}"
872,1,145,0,0.0649014,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin ^3(c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin ^3(c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{18, 2748, 3767, 3768, 3770}"
873,1,148,0,0.0556814,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{A x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
874,1,107,0,0.0232636,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]])","A",2,2,33,0.06061,1,"{17, 2734}"
875,1,65,0,0.0136778,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
876,1,66,0,0.0266115,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]])","A",3,3,33,0.09091,1,"{17, 2735, 3770}"
877,1,74,0,0.0415016,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)),x]","\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",5,5,33,0.1515,1,"{18, 2748, 3767, 8, 3770}"
878,1,116,0,0.0561972,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{A \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",6,6,33,0.1818,1,"{18, 2748, 3768, 3770, 3767, 8}"
879,1,157,0,0.0644936,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{A \sin ^3(c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}","\frac{A \sin ^3(c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{18, 2748, 3767, 3768, 3770}"
880,1,148,0,0.0569816,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{A x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{17, 2748, 2635, 8, 2633}"
881,1,107,0,0.0240382,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]])","A",2,2,33,0.06061,1,"{17, 2734}"
882,1,65,0,0.0136397,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(A*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",3,2,33,0.06061,1,"{17, 2637}"
883,1,66,0,0.0274445,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}",1,"(B*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",3,3,33,0.09091,1,"{17, 2735, 3770}"
884,1,74,0,0.0413233,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",5,5,33,0.1515,1,"{17, 2748, 3767, 8, 3770}"
885,1,116,0,0.0560446,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)),x]","\frac{A \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",6,6,33,0.1818,1,"{18, 2748, 3768, 3770, 3767, 8}"
886,1,157,0,0.0642757,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{A \sin ^3(c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin ^3(c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])","A",6,5,33,0.1515,1,"{18, 2748, 3767, 3768, 3770}"
887,1,119,0,0.07488,"\int \cos ^2(c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^4 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^4*d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
888,1,119,0,0.0742312,"\int \cos (c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^3*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
889,1,119,0,0.0606157,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^2*d*Sqrt[Sin[c + d*x]^2])","A",3,2,23,0.08696,1,"{2748, 2643}"
890,1,114,0,0.0789069,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
891,1,112,0,0.0897038,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}",1,"(3*A*b*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
892,1,117,0,0.0909964,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(3*A*b^2*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
893,1,119,0,0.0736791,"\int \cos ^2(c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{16/3} \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)}{16 b^4 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{16/3} \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)}{16 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(16/3)*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2]*Sin[c + d*x])/(16*b^4*d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
894,1,119,0,0.074004,"\int \cos (c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
895,1,119,0,0.0641907,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Int[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])","A",3,2,23,0.08696,1,"{2748, 2643}"
896,1,116,0,0.0801155,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
897,1,112,0,0.0956602,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*b*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
898,1,115,0,0.0982533,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}",1,"(3*A*b^2*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
899,1,119,0,0.075725,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^4 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^4*d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
900,1,119,0,0.0700779,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
901,1,117,0,0.0672679,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Int[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])","A",3,2,23,0.08696,1,"{2748, 2643}"
902,1,114,0,0.0850318,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(2/3),x]","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}",1,"(3*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
903,1,114,0,0.0973742,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(2/3),x]","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(3*A*b*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
904,1,117,0,0.0997625,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(2/3),x]","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{8/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{8/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}",1,"(3*A*b^2*Hypergeometric2F1[-4/3, 1/2, -1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Cos[c + d*x])^(8/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
905,1,119,0,0.0734252,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^4 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^4*d*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
906,1,119,0,0.0725229,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Int[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
907,1,117,0,0.0635625,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Int[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(3*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])","A",3,2,23,0.08696,1,"{2748, 2643}"
908,1,114,0,0.0839064,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(3*A*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
909,1,114,0,0.0971097,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*A*b*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
910,1,117,0,0.0980307,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{10/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{10/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}",1,"(3*A*b^2*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Cos[c + d*x])^(10/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{16, 2748, 2643}"
911,1,157,0,0.0842588,"\int \cos ^m(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{A \sin (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) \cos ^{m+2}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)}{d (m+n+2) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) \cos ^{m+2}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)}{d (m+n+2) \sqrt{\sin ^2(c+d x)}}",1,"-((A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])) - (B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{20, 2748, 2643}"
912,1,141,0,0.0991279,"\int \cos ^2(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}",1,"-((A*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(4 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^4*d*(4 + n)*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
913,1,141,0,0.0948405,"\int \cos (c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}",1,"-((A*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])","A",4,3,27,0.1111,1,"{16, 2748, 2643}"
914,1,141,0,0.0796793,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}",1,"-((A*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])","A",3,2,21,0.09524,1,"{2748, 2643}"
915,1,132,0,0.0937742,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx","Int[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}",1,"-((A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])","A",4,3,27,0.1111,1,"{16, 2748, 2643}"
916,1,131,0,0.1141235,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Int[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{A b \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}","\frac{A b \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}",1,"(A*b*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
917,1,139,0,0.1173737,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Int[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{A b^2 \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}+\frac{b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}","\frac{A b^2 \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}+\frac{b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}",1,"(A*b^2*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) + (b*B*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
918,1,141,0,0.1194448,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Int[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{A b^3 \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)}{d (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}","\frac{A b^3 \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)}{d (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}",1,"(A*b^3*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]) + (b^2*B*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])","A",4,3,29,0.1034,1,"{16, 2748, 2643}"
919,1,163,0,0.0946512,"\int \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)}{d (2 n+9) \sqrt{\sin ^2(c+d x)}}","-\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)}{d (2 n+9) \sqrt{\sin ^2(c+d x)}}",1,"(-2*A*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(9/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(9 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
920,1,163,0,0.0912102,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}","-\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}",1,"(-2*A*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
921,1,163,0,0.0864296,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}","-\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}",1,"(-2*A*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
922,1,163,0,0.0845629,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Int[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","-\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}","-\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}",1,"(-2*A*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
923,1,163,0,0.0878826,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}",1,"(2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]) - (2*B*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
924,1,163,0,0.0960241,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
925,1,163,0,0.0910064,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
926,1,163,0,0.0906357,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Int[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)}{d (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)}{d (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-7 + 2*n)/4, (-3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 - 2*n)*Cos[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
927,1,169,0,0.099053,"\int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*b*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (10 + 3*m)/6, (16 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
928,1,167,0,0.0945796,"\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (8 + 3*m)/6, (14 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
929,1,167,0,0.0863866,"\int \cos ^m(c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Int[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}",1,"(-3*A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
930,1,167,0,0.0874664,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{\sqrt[3]{b \cos (c+d x)}} \, dx","Int[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*A*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
931,1,167,0,0.0880054,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Int[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)}{d (3 m+1) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)}{d (3 m+1) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(-3*A*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"
932,1,171,0,0.0974893,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Int[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","\frac{3 A \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)}{b d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","\frac{3 A \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)}{b d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(3*A*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",4,3,31,0.09677,1,"{20, 2748, 2643}"