1,1,85,0,0.058626,"\int \sec ^4(c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sec[c + d*x]),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,"{3787, 3767, 3768, 3770}"
2,1,63,0,0.0457791,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x]),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,"{3787, 3768, 3770, 3767}"
3,1,47,0,0.0421372,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x]),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,"{3787, 3767, 8, 3768, 3770}"
4,1,24,0,0.0233012,"\int \sec (c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x]),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",4,4,17,0.2353,1,"{3787, 3770, 3767, 8}"
5,1,16,0,0.0072979,"\int (a+a \sec (c+d x)) \, dx","Int[a + a*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",2,1,10,0.1000,1,"{3770}"
6,1,15,0,0.0192023,"\int \cos (c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sec[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",3,3,17,0.1765,1,"{3787, 2637, 8}"
7,1,38,0,0.0346483,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[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",4,4,19,0.2105,1,"{3787, 2635, 8, 2637}"
8,1,54,0,0.0404894,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[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,"{3787, 2633, 2635, 8}"
9,1,76,0,0.0550533,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[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,"{3787, 2635, 8, 2633}"
10,1,122,0,0.0946708,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","\frac{3 a^2 \tan ^3(c+d x)}{5 d}+\frac{9 a^2 \tan (c+d x)}{5 d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}","\frac{3 a^2 \tan ^3(c+d x)}{5 d}+\frac{9 a^2 \tan (c+d x)}{5 d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}",1,"(3*a^2*ArcTanh[Sin[c + d*x]])/(4*d) + (9*a^2*Tan[c + d*x])/(5*d) + (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (a^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (3*a^2*Tan[c + d*x]^3)/(5*d)","A",7,5,21,0.2381,1,"{3788, 3768, 3770, 4046, 3767}"
11,1,96,0,0.08384,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2,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",6,5,21,0.2381,1,"{3788, 3767, 4046, 3768, 3770}"
12,1,74,0,0.0808184,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","\frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}","\frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/d + (5*a^2*Tan[c + d*x])/(3*d) + (a^2*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,"{3788, 3768, 3770, 4046, 3767, 8}"
13,1,54,0,0.0470463,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^2,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",5,5,19,0.2632,1,"{3788, 3767, 8, 4046, 3770}"
14,1,34,0,0.0237475,"\int (a+a \sec (c+d x))^2 \, dx","Int[(a + a*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",4,4,12,0.3333,1,"{3773, 3770, 3767, 8}"
15,1,34,0,0.0508976,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^2,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",4,4,19,0.2105,1,"{3788, 8, 4045, 3770}"
16,1,45,0,0.0602976,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[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",4,4,21,0.1905,1,"{3788, 2637, 4045, 8}"
17,1,57,0,0.0821145,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","-\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","-\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 + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d)","A",6,5,21,0.2381,1,"{3788, 2635, 8, 4044, 3013}"
18,1,87,0,0.0793729,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[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",6,5,21,0.2381,1,"{3788, 2633, 4045, 2635, 8}"
19,1,103,0,0.1098046,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[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",8,6,21,0.2857,1,"{3788, 2635, 8, 4044, 3013, 373}"
20,1,114,0,0.1252914,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3,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,"{3791, 3768, 3770, 3767}"
21,1,93,0,0.1139875,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3,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,"{3791, 3767, 8, 3768, 3770}"
22,1,72,0,0.0752543,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^3,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",9,5,19,0.2632,1,"{3791, 3770, 3767, 8, 3768}"
23,1,66,0,0.0473646,"\int (a+a \sec (c+d x))^3 \, dx","Int[(a + a*Sec[c + d*x])^3,x]","\frac{5 a^3 \tan (c+d x)}{2 d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+a^3 x","\frac{5 a^3 \tan (c+d x)}{2 d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+a^3 x",1,"a^3*x + (7*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*Tan[c + d*x])/(2*d) + ((a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",5,5,12,0.4167,1,"{3775, 3914, 3767, 8, 3770}"
24,1,48,0,0.0579294,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^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","\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",6,5,19,0.2632,1,"{3791, 2637, 3770, 3767, 8}"
25,1,59,0,0.066875,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3,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",6,5,21,0.2381,1,"{3791, 2637, 2635, 8, 3770}"
26,1,63,0,0.0746903,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[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",7,5,21,0.2381,1,"{3791, 2637, 2635, 8, 2633}"
27,1,85,0,0.0972659,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","-\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}","-\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 + (4*a^3*Sin[c + d*x])/d + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/d","A",10,5,21,0.2381,1,"{3791, 2637, 2635, 8, 2633}"
28,1,105,0,0.1118694,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[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,"{3791, 2635, 8, 2633}"
29,1,129,0,0.1344709,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[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,"{3791, 2633, 2635, 8}"
30,1,136,0,0.1717275,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^4,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,"{3791, 3768, 3770, 3767}"
31,1,111,0,0.135141,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4,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,"{3791, 3767, 8, 3768, 3770}"
32,1,96,0,0.1092646,"\int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^4,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",12,5,19,0.2632,1,"{3791, 3770, 3767, 8, 3768}"
33,1,91,0,0.0898229,"\int (a+a \sec (c+d x))^4 \, dx","Int[(a + a*Sec[c + d*x])^4,x]","\frac{5 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 d}+\frac{4 \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{3 d}+a^4 x","\frac{5 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 d}+\frac{4 \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{3 d}+a^4 x",1,"a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (5*a^4*Tan[c + d*x])/d + ((a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + (4*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(3*d)","A",6,6,12,0.5000,1,"{3775, 3917, 3914, 3767, 8, 3770}"
34,1,73,0,0.0757867,"\int \cos (c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^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","\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",8,6,19,0.3158,1,"{3791, 2637, 3770, 3767, 8, 3768}"
35,1,73,0,0.078554,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4,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,"{3791, 2637, 2635, 8, 3770, 3767}"
36,1,73,0,0.0815262,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^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","-\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",8,6,21,0.2857,1,"{3791, 2637, 2635, 8, 2633, 3770}"
37,1,87,0,0.098184,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[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",10,5,21,0.2381,1,"{3791, 2637, 2635, 8, 2633}"
38,1,102,0,0.1138799,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4,x]","\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}","\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 + (8*a^4*Sin[c + d*x])/d + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d - (8*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^5)/(5*d)","A",12,5,21,0.2381,1,"{3791, 2637, 2635, 8, 2633}"
39,1,127,0,0.1453003,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[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,"{3791, 2635, 8, 2633}"
40,1,147,0,0.1548267,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4,x]","-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{9 a^4 \sin ^5(c+d x)}{5 d}-\frac{16 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{11 a^4 x}{4}","-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{9 a^4 \sin ^5(c+d x)}{5 d}-\frac{16 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{11 a^4 x}{4}",1,"(11*a^4*x)/4 + (8*a^4*Sin[c + d*x])/d + (11*a^4*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (11*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (2*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(3*d) - (16*a^4*Sin[c + d*x]^3)/(3*d) + (9*a^4*Sin[c + d*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/(7*d)","A",15,4,21,0.1905,1,"{3791, 2633, 2635, 8}"
41,1,156,0,0.1978649,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^5,x]","\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d}","\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d}",1,"(93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (16*a^5*Tan[c + d*x])/d + (93*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (85*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (28*a^5*Tan[c + d*x]^3)/(3*d) + (13*a^5*Tan[c + d*x]^5)/(5*d) + (a^5*Tan[c + d*x]^7)/(7*d)","A",17,4,21,0.1905,1,"{3791, 3768, 3770, 3767}"
42,1,103,0,0.098244,"\int \frac{\sec ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[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{\tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}","\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{\tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}",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]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + (4*Tan[c + d*x]^3)/(3*a*d)","A",6,5,21,0.2381,1,"{3818, 3787, 3768, 3770, 3767}"
43,1,85,0,0.0927384,"\int \frac{\sec ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x]),x]","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}",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]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,21,0.2857,1,"{3818, 3787, 3767, 8, 3768, 3770}"
44,1,51,0,0.1055284,"\int \frac{\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{\tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}+\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{\tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}+\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"-(ArcTanh[Sin[c + d*x]]/(a*d)) + Tan[c + d*x]/(a*d) + Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))","A",4,4,21,0.1905,1,"{3790, 3789, 3770, 3794}"
45,1,38,0,0.067653,"\int \frac{\sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"ArcTanh[Sin[c + d*x]]/(a*d) - Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))","A",3,3,21,0.1429,1,"{3789, 3770, 3794}"
46,1,22,0,0.0238029,"\int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))","A",1,1,19,0.05263,1,"{3794}"
47,1,29,0,0.0135317,"\int \frac{1}{a+a \sec (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(-1),x]","\frac{x}{a}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{x}{a}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"x/a - Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))","A",2,2,12,0.1667,1,"{3777, 8}"
48,1,44,0,0.0571631,"\int \frac{\cos (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x}{a}","\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x}{a}",1,"-(x/a) + (2*Sin[c + d*x])/(a*d) - Sin[c + d*x]/(d*(a + a*Sec[c + d*x]))","A",4,4,19,0.2105,1,"{3819, 3787, 2637, 8}"
49,1,74,0,0.0816637,"\int \frac{\cos ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x]),x]","-\frac{2 \sin (c+d x)}{a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x}{2 a}","-\frac{2 \sin (c+d x)}{a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\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]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3819, 3787, 2635, 8, 2637}"
50,1,94,0,0.0903989,"\int \frac{\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + a*Sec[c + d*x]),x]","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 x}{2 a}","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\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]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - (4*Sin[c + d*x]^3)/(3*a*d)","A",6,5,21,0.2381,1,"{3819, 3787, 2633, 2635, 8}"
51,1,118,0,0.1012673,"\int \frac{\cos ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}+\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{\sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{15 x}{8 a}","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}+\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{\sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\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]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + (4*Sin[c + d*x]^3)/(3*a*d)","A",7,5,21,0.2381,1,"{3819, 3787, 2635, 8, 2633}"
52,1,123,0,0.1794145,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[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{8 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (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{8 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (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]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,7,21,0.3333,1,"{3816, 4019, 3787, 3767, 8, 3768, 3770}"
53,1,89,0,0.1551679,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{4 \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 (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{4 \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 (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*ArcTanh[Sin[c + d*x]])/(a^2*d) + (4*Tan[c + d*x])/(3*a^2*d) + (2*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,21,0.2857,1,"{3816, 4008, 3787, 3770, 3767, 8}"
54,1,66,0,0.1181351,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(a^2*d) - (5*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)","A",4,4,21,0.1905,1,"{3799, 3998, 3770, 3794}"
55,1,55,0,0.0656275,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{2 \tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{2 \tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(3*d*(a^2 + a^2*Sec[c + d*x]))","A",2,2,21,0.09524,1,"{3797, 3794}"
56,1,55,0,0.0493647,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{\tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{\tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2) + Tan[c + d*x]/(3*d*(a^2 + a^2*Sec[c + d*x]))","A",2,2,19,0.1053,1,"{3796, 3794}"
57,1,57,0,0.0694143,"\int \frac{1}{(a+a \sec (c+d x))^2} \, dx","Int[(a + a*Sec[c + d*x])^(-2),x]","-\frac{4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x}{a^2}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x}{a^2}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"x/a^2 - (4*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)","A",3,3,12,0.2500,1,"{3777, 3919, 3794}"
58,1,72,0,0.1293726,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}","\frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*x)/a^2 + (10*Sin[c + d*x])/(3*a^2*d) - (2*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)","A",5,5,19,0.2632,1,"{3817, 4020, 3787, 2637, 8}"
59,1,110,0,0.1750239,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","-\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{8 \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}","-\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{8 \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos (c+d x)}{3 d (a \sec (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]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,21,0.2857,1,"{3817, 4020, 3787, 2635, 8, 2637}"
60,1,124,0,0.1904354,"\int \frac{\cos ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + a*Sec[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{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (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{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (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]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*Sin[c + d*x]^3)/(a^2*d)","A",7,6,21,0.2857,1,"{3817, 4020, 3787, 2633, 2635, 8}"
61,1,162,0,0.2921551,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^6/(a + a*Sec[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{76 \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{\tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{11 \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}","-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{76 \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{\tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{11 \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",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]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (11*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (76*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,21,0.3333,1,"{3816, 4019, 3787, 3767, 8, 3768, 3770}"
62,1,128,0,0.2647082,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{9 \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 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{3 \tan (c+d x) \sec ^2(c+d x)}{5 a d (a \sec (c+d x)+a)^2}","\frac{9 \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 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{3 \tan (c+d x) \sec ^2(c+d x)}{5 a d (a \sec (c+d x)+a)^2}",1,"(-3*ArcTanh[Sin[c + d*x]])/(a^3*d) + (9*Tan[c + d*x])/(5*a^3*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (3*Sec[c + d*x]^2*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) + (3*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",7,7,21,0.3333,1,"{3816, 4019, 4008, 3787, 3770, 3767, 8}"
63,1,105,0,0.221938,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{29 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{29 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(a^3*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (7*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (29*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3816, 4008, 3998, 3770, 3794}"
64,1,83,0,0.1233833,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","\frac{7 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}","\frac{7 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (7*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,21,0.1429,1,"{3799, 4000, 3794}"
65,1,83,0,0.0967388,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{\tan (c+d x)}{5 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\tan (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}","\frac{\tan (c+d x)}{5 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\tan (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"-Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) + Tan[c + d*x]/(5*a*d*(a + a*Sec[c + d*x])^2) + Tan[c + d*x]/(5*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,21,0.1429,1,"{3797, 3796, 3794}"
66,1,83,0,0.080896,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{2 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{2 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}","\frac{2 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{2 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,2,19,0.1053,1,"{3796, 3794}"
67,1,88,0,0.1123227,"\int \frac{1}{(a+a \sec (c+d x))^3} \, dx","Int[(a + a*Sec[c + d*x])^(-3),x]","-\frac{22 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}","-\frac{22 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"x/a^3 - Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (7*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (22*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,4,12,0.3333,1,"{3777, 3922, 3919, 3794}"
68,1,103,0,0.220573,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{24 \sin (c+d x)}{5 a^3 d}-\frac{3 \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}","\frac{24 \sin (c+d x)}{5 a^3 d}-\frac{3 \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(-3*x)/a^3 + (24*Sin[c + d*x])/(5*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (3*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (3*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",6,5,19,0.2632,1,"{3817, 4020, 3787, 2637, 8}"
69,1,147,0,0.2900272,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","-\frac{152 \sin (c+d x)}{15 a^3 d}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{76 \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 x}{2 a^3}-\frac{11 \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}","-\frac{152 \sin (c+d x)}{15 a^3 d}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{76 \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 x}{2 a^3}-\frac{11 \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",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]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (11*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (76*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",7,6,21,0.2857,1,"{3817, 4020, 3787, 2635, 8, 2637}"
70,1,193,0,0.3933181,"\int \frac{\sec ^7(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^7/(a + a*Sec[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{43 \tan (c+d x) \sec ^3(c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}-\frac{288 \tan (c+d x) \sec ^2(c+d x)}{35 a^4 d (\sec (c+d x)+1)}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}","-\frac{576 \tan (c+d x)}{35 a^4 d}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{43 \tan (c+d x) \sec ^3(c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}-\frac{288 \tan (c+d x) \sec ^2(c+d x)}{35 a^4 d (\sec (c+d x)+1)}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}",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]^3*Tan[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])^2) - (288*Sec[c + d*x]^2*Tan[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",9,7,21,0.3333,1,"{3816, 4019, 3787, 3767, 8, 3768, 3770}"
71,1,159,0,0.3687136,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^6/(a + a*Sec[c + d*x])^4,x]","\frac{244 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}","\frac{244 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(-4*ArcTanh[Sin[c + d*x]])/(a^4*d) + (244*Tan[c + d*x])/(105*a^4*d) - (88*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (12*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",8,7,21,0.3333,1,"{3816, 4019, 4008, 3787, 3770, 3767, 8}"
72,1,136,0,0.3232205,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^4,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{43 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}+\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}-\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^2(c+d x)}{7 a d (a \sec (c+d x)+a)^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{43 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}+\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}-\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^2(c+d x)}{7 a d (a \sec (c+d x)+a)^3}",1,"ArcTanh[Sin[c + d*x]]/(a^4*d) + (11*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])^2) - (43*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Sec[c + d*x]^2*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^3)","A",6,6,21,0.2857,1,"{3816, 4019, 4008, 3998, 3770, 3794}"
73,1,120,0,0.1686549,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^4,x]","\frac{\tan (c+d x)}{5 d \left(a^4 \sec (c+d x)+a^4\right)}-\frac{8 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}","\frac{\tan (c+d x)}{5 d \left(a^4 \sec (c+d x)+a^4\right)}-\frac{8 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(35*d*(a^2 + a^2*Sec[c + d*x])^2) + Tan[c + d*x]/(5*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,21,0.1905,1,"{3810, 3799, 4000, 3794}"
74,1,112,0,0.1541207,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^4,x]","\frac{13 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{13 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}-\frac{11 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}","\frac{13 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{13 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}-\frac{11 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (11*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (13*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (13*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,21,0.1905,1,"{3799, 4000, 3796, 3794}"
75,1,112,0,0.125312,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^4,x]","\frac{8 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{8 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}","\frac{8 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{8 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"-Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) + (4*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (8*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (8*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,3,21,0.1429,1,"{3797, 3796, 3794}"
76,1,112,0,0.1108479,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^4,x]","\frac{2 \tan (c+d x)}{35 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{2 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}","\frac{2 \tan (c+d x)}{35 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{2 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) + (3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(35*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(35*d*(a^4 + a^4*Sec[c + d*x]))","A",4,2,19,0.1053,1,"{3796, 3794}"
77,1,111,0,0.1602959,"\int \frac{1}{(a+a \sec (c+d x))^4} \, dx","Int[(a + a*Sec[c + d*x])^(-4),x]","-\frac{32 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}-\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}+\frac{x}{a^4}-\frac{2 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}","-\frac{32 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}-\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}+\frac{x}{a^4}-\frac{2 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"x/a^4 - (11*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])^2) - (32*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])) - Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (2*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^3)","A",5,4,12,0.3333,1,"{3777, 3922, 3919, 3794}"
78,1,126,0,0.3042294,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^4,x]","\frac{664 \sin (c+d x)}{105 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{88 \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 x}{a^4}-\frac{12 \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}","\frac{664 \sin (c+d x)}{105 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{88 \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 x}{a^4}-\frac{12 \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(-4*x)/a^4 + (664*Sin[c + d*x])/(105*a^4*d) - (88*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (12*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",7,5,19,0.2632,1,"{3817, 4020, 3787, 2637, 8}"
79,1,176,0,0.3929533,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^4,x]","-\frac{576 \sin (c+d x)}{35 a^4 d}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{288 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)}-\frac{43 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}+\frac{21 x}{2 a^4}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}","-\frac{576 \sin (c+d x)}{35 a^4 d}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{288 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)}-\frac{43 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}+\frac{21 x}{2 a^4}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}",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]*Sin[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])^2) - (288*Cos[c + d*x]*Sin[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",8,6,21,0.2857,1,"{3817, 4020, 3787, 2635, 8, 2637}"
80,1,200,0,0.4802324,"\int \frac{\sec ^7(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^7/(a + a*Sec[c + d*x])^5,x]","\frac{181 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{29 \tan (c+d x) \sec ^3(c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{67 \tan (c+d x) \sec ^2(c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}+\frac{5 \tan (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{\tan (c+d x) \sec ^5(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{5 \tan (c+d x) \sec ^4(c+d x)}{21 a d (a \sec (c+d x)+a)^4}","\frac{181 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{29 \tan (c+d x) \sec ^3(c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{67 \tan (c+d x) \sec ^2(c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}+\frac{5 \tan (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{\tan (c+d x) \sec ^5(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{5 \tan (c+d x) \sec ^4(c+d x)}{21 a d (a \sec (c+d x)+a)^4}",1,"(-5*ArcTanh[Sin[c + d*x]])/(a^5*d) + (181*Tan[c + d*x])/(63*a^5*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sec[c + d*x]^4*Tan[c + d*x])/(21*a*d*(a + a*Sec[c + d*x])^4) - (29*Sec[c + d*x]^3*Tan[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sec[c + d*x]^2*Tan[c + d*x])/(63*a^3*d*(a + a*Sec[c + d*x])^2) + (5*Tan[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))","A",9,7,21,0.3333,1,"{3816, 4019, 4008, 3787, 3770, 3767, 8}"
81,1,177,0,0.4272646,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^6/(a + a*Sec[c + d*x])^5,x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{34 \tan (c+d x) \sec ^2(c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac{661 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{13 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{34 \tan (c+d x) \sec ^2(c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac{661 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{13 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"ArcTanh[Sin[c + d*x]]/(a^5*d) - (Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (13*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) + (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (661*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))","A",7,6,21,0.2857,1,"{3816, 4019, 4008, 3998, 3770, 3794}"
82,1,159,0,0.2159525,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^5,x]","\frac{4 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{32 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{4 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}","\frac{4 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{32 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{4 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"(Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) + (4*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + (4*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (32*Tan[c + d*x])/(315*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (4*Tan[c + d*x])/(45*d*(a^5 + a^5*Sec[c + d*x]))","A",5,4,21,0.1905,1,"{3810, 3799, 4000, 3794}"
83,1,159,0,0.2148033,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^5,x]","\frac{\tan (c+d x)}{9 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{8 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}","\frac{\tan (c+d x)}{9 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{8 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"-(Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(21*a^2*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(63*a*d*(a^2 + a^2*Sec[c + d*x])^2) + Tan[c + d*x]/(9*d*(a^5 + a^5*Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3811, 3810, 3799, 4000, 3794}"
84,1,139,0,0.1820881,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^5,x]","\frac{2 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac{2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}","\frac{2 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac{2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (2*Tan[c + d*x])/(9*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(15*a^2*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(45*a^3*d*(a + a*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(45*d*(a^5 + a^5*Sec[c + d*x]))","A",5,4,21,0.1905,1,"{3799, 4000, 3796, 3794}"
85,1,143,0,0.1573239,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]","\frac{2 \tan (c+d x)}{63 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}+\frac{5 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}","\frac{2 \tan (c+d x)}{63 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}+\frac{5 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"-Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) + (5*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(21*a^2*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(63*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(63*d*(a^5 + a^5*Sec[c + d*x]))","A",5,3,21,0.1429,1,"{3797, 3796, 3794}"
86,1,143,0,0.1423802,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^5,x]","\frac{8 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{8 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{4 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}","\frac{8 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{8 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{4 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) + (4*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + (4*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) + (8*Tan[c + d*x])/(315*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (8*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))","A",5,2,19,0.1053,1,"{3796, 3794}"
87,1,144,0,0.2073062,"\int \frac{1}{(a+a \sec (c+d x))^5} \, dx","Int[(a + a*Sec[c + d*x])^(-5),x]","-\frac{488 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{x}{a^5}-\frac{13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}","-\frac{488 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{x}{a^5}-\frac{13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"x/a^5 - Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (13*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (488*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))","A",6,4,12,0.3333,1,"{3777, 3922, 3919, 3794}"
88,1,159,0,0.3972075,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^5,x]","\frac{496 \sin (c+d x)}{63 a^5 d}-\frac{5 \sin (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{5 x}{a^5}-\frac{5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5}","\frac{496 \sin (c+d x)}{63 a^5 d}-\frac{5 \sin (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{5 x}{a^5}-\frac{5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(-5*x)/a^5 + (496*Sin[c + d*x])/(63*a^5*d) - Sin[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sin[c + d*x])/(21*a*d*(a + a*Sec[c + d*x])^4) - (29*Sin[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sin[c + d*x])/(63*a^3*d*(a + a*Sec[c + d*x])^2) - (5*Sin[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))","A",8,5,19,0.2632,1,"{3817, 4020, 3787, 2637, 8}"
89,1,215,0,0.5093668,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]","-\frac{7664 \sin (c+d x)}{315 a^5 d}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac{3832 \sin (c+d x) \cos (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}+\frac{31 x}{2 a^5}-\frac{17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}","-\frac{7664 \sin (c+d x)}{315 a^5 d}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac{3832 \sin (c+d x) \cos (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}+\frac{31 x}{2 a^5}-\frac{17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}",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]*Sin[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (17*Cos[c + d*x]*Sin[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (28*Cos[c + d*x]*Sin[c + d*x])/(45*a^2*d*(a + a*Sec[c + d*x])^3) - (577*Cos[c + d*x]*Sin[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (3832*Cos[c + d*x]*Sin[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))","A",9,6,21,0.2857,1,"{3817, 4020, 3787, 2635, 8, 2637}"
90,1,122,0,0.2068566,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{35 d}+\frac{4 a \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}","\frac{2 a \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{35 d}+\frac{4 a \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(4*a*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (8*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(35*d) + (12*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)","A",4,4,23,0.1739,1,"{3803, 3800, 4001, 3792}"
91,1,86,0,0.1520131,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{14 a \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{14 a \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}",1,"(14*a*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)","A",3,3,23,0.1304,1,"{3800, 4001, 3792}"
92,1,56,0,0.0826257,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",2,2,23,0.08696,1,"{3798, 3792}"
93,1,26,0,0.0294322,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",1,1,21,0.04762,1,"{3792}"
94,1,37,0,0.0226566,"\int \sqrt{a+a \sec (c+d x)} \, dx","Int[Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d","A",2,2,14,0.1429,1,"{3774, 203}"
95,1,62,0,0.0628327,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,21,0.1429,1,"{3805, 3774, 203}"
96,1,102,0,0.1182282,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","\frac{3 a \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}","\frac{3 a \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(3*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (3*a*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",4,3,23,0.1304,1,"{3805, 3774, 203}"
97,1,138,0,0.1778554,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{5 a \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{5 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}","\frac{5 a \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{5 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(5*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (5*a*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,3,23,0.1304,1,"{3805, 3774, 203}"
98,1,174,0,0.2350994,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\frac{35 a \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{35 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{35 a \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}","\frac{35 a \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{35 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{35 a \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}",1,"(35*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (35*a*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (35*a*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (7*a*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])","A",6,3,23,0.1304,1,"{3805, 3774, 203}"
99,1,162,0,0.2752719,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{68 a^2 \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{68 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}","\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{68 a^2 \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{68 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}",1,"(68*a^2*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (34*a^2*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (136*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (68*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)","A",6,6,23,0.2609,1,"{3814, 21, 3803, 3800, 4001, 3792}"
100,1,116,0,0.1939412,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","\frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}","\frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}",1,"(152*a^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (38*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) - (4*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)","A",4,4,23,0.1739,1,"{3800, 4001, 3793, 3792}"
101,1,86,0,0.1199567,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","\frac{8 a^2 \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{8 a^2 \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(8*a^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",3,3,23,0.1304,1,"{3798, 3793, 3792}"
102,1,59,0,0.0614329,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{8 a^2 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}","\frac{8 a^2 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(8*a^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",2,2,21,0.09524,1,"{3793, 3792}"
103,1,66,0,0.0368724,"\int (a+a \sec (c+d x))^{3/2} \, dx","Int[(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,14,0.2857,1,"{3775, 21, 3774, 203}"
104,1,65,0,0.1181843,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a^2 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(3*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,21,0.2381,1,"{3814, 21, 3805, 3774, 203}"
105,1,106,0,0.1266022,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","\frac{7 a^2 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}","\frac{7 a^2 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(7*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (7*a^2*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3813, 21, 3805, 3774, 203}"
106,1,144,0,0.1865153,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}","\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(11*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (11*a^2*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,23,0.2174,1,"{3813, 21, 3805, 3774, 203}"
107,1,203,0,0.373957,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}+\frac{46 a^3 \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}-\frac{568 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{284 a^3 \tan (c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{284 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{231 d}","\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}+\frac{46 a^3 \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}-\frac{568 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{284 a^3 \tan (c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{284 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{231 d}",1,"(284*a^3*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (710*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (46*a^3*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (568*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(693*d) + (2*a^2*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (284*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(231*d)","A",6,6,23,0.2609,1,"{3814, 4016, 3803, 3800, 4001, 3792}"
108,1,146,0,0.2304717,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","\frac{208 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{832 a^3 \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}","\frac{208 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{832 a^3 \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}",1,"(832*a^3*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (208*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (26*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) - (4*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)","A",5,4,23,0.1739,1,"{3800, 4001, 3793, 3792}"
109,1,116,0,0.1551429,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}","\frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(64*a^3*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",4,3,23,0.1304,1,"{3798, 3793, 3792}"
110,1,89,0,0.0949433,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 a^3 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{64 a^3 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(64*a^3*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",3,2,21,0.09524,1,"{3793, 3792}"
111,1,98,0,0.1022338,"\int (a+a \sec (c+d x))^{5/2} \, dx","Int[(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (14*a^3*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",5,5,14,0.3571,1,"{3775, 3915, 3774, 203, 3792}"
112,1,94,0,0.1570408,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{a^3 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{5 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","-\frac{a^3 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{5 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(5*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,21,0.1905,1,"{3814, 4015, 3774, 203}"
113,1,106,0,0.1643924,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{19 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}","\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{19 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(19*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,23,0.1739,1,"{3813, 4015, 3774, 203}"
114,1,144,0,0.2345466,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","\frac{25 a^3 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{25 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}","\frac{25 a^3 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{25 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(25*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (25*a^3*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,23,0.2174,1,"{3813, 4015, 3805, 3774, 203}"
115,1,182,0,0.294107,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","\frac{163 a^3 \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{163 a^3 \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}","\frac{163 a^3 \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{163 a^3 \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}",1,"(163*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (163*a^3*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (17*a^3*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,23,0.2174,1,"{3813, 4015, 3805, 3774, 203}"
116,1,27,0,0.0298956,"\int \sec (c+d x) \sqrt{a-a \sec (c+d x)} \, dx","Int[Sec[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]","-\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}}","-\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}}",1,"(-2*a*Tan[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])","A",1,1,22,0.04545,1,"{3792}"
117,1,38,0,0.0219381,"\int \sqrt{a-a \sec (c+d x)} \, dx","Int[Sqrt[a - a*Sec[c + d*x]],x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d","A",2,2,15,0.1333,1,"{3774, 203}"
118,1,65,0,0.0638623,"\int \cos (c+d x) \sqrt{a-a \sec (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]","\frac{a \sin (c+d x)}{d \sqrt{a-a \sec (c+d x)}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}","\frac{a \sin (c+d x)}{d \sqrt{a-a \sec (c+d x)}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d) + (a*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])","A",3,3,22,0.1364,1,"{3805, 3774, 203}"
119,1,140,0,0.2758992,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}+\frac{28 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}","\frac{2 \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}+\frac{28 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (28*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)","A",5,5,23,0.2174,1,"{3822, 4010, 4001, 3795, 203}"
120,1,104,0,0.157705,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}-\frac{4 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}-\frac{4 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)","A",4,4,23,0.1739,1,"{3800, 4001, 3795, 203}"
121,1,73,0,0.0876906,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,23,0.1304,1,"{3798, 3795, 203}"
122,1,46,0,0.0359379,"\int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",2,2,21,0.09524,1,"{3795, 203}"
123,1,85,0,0.0664476,"\int \frac{1}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[1/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,4,14,0.2857,1,"{3776, 3774, 203, 3795}"
124,1,108,0,0.1787319,"\int \frac{\cos (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-(ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,21,0.2381,1,"{3823, 3904, 3887, 481, 203}"
125,1,147,0,0.2483742,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}","-\frac{\sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(7*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - Sin[c + d*x]/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,23,0.2609,1,"{3823, 4022, 3920, 3774, 203, 3795}"
126,1,183,0,0.4246417,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{15 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{13 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{10 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{9 \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{31 \tan (c+d x)}{5 a d \sqrt{a \sec (c+d x)+a}}","-\frac{15 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{13 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{10 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{9 \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{31 \tan (c+d x)}{5 a d \sqrt{a \sec (c+d x)+a}}",1,"(-15*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (31*Tan[c + d*x])/(5*a*d*Sqrt[a + a*Sec[c + d*x]]) + (9*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - (13*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(10*a^2*d)","A",6,6,23,0.2609,1,"{3816, 4021, 4010, 4001, 3795, 203}"
127,1,145,0,0.2883222,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}","\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"(11*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (13*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + (7*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)","A",5,5,23,0.2174,1,"{3816, 4010, 4001, 3795, 203}"
128,1,105,0,0.1681732,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(-7*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,23,0.1739,1,"{3799, 4001, 3795, 203}"
129,1,77,0,0.0973793,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,23,0.1304,1,"{3797, 3795, 203}"
130,1,77,0,0.0720289,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,21,0.1429,1,"{3796, 3795, 203}"
131,1,114,0,0.1157111,"\int \frac{1}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])^(-3/2),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,5,14,0.3571,1,"{3777, 3920, 3774, 203, 3795}"
132,1,144,0,0.2575083,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(-3*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (9*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,21,0.2857,1,"{3817, 4022, 3920, 3774, 203, 3795}"
133,1,185,0,0.3906973,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{13 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x) \cos (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{13 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x) \cos (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(19*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (13*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (7*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,23,0.2609,1,"{3817, 4022, 3920, 3774, 203, 3795}"
134,1,183,0,0.4239824,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{163 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{95 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{197 \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","\frac{163 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{95 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{197 \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(163*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (17*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (197*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + (95*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)","A",6,6,23,0.2609,1,"{3816, 4019, 4010, 4001, 3795, 203}"
135,1,145,0,0.3088285,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{75 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{9 \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","-\frac{75 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{9 \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-75*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (13*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (9*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3816, 4008, 4001, 3795, 203}"
136,1,107,0,0.1755997,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(19*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (13*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{3799, 4000, 3795, 203}"
137,1,107,0,0.1367722,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(5*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (5*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{3797, 3796, 3795, 203}"
138,1,107,0,0.1133949,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,3,21,0.1429,1,"{3796, 3795, 203}"
139,1,144,0,0.1764683,"\int \frac{1}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])^(-5/2),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{11 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{11 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (11*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,6,14,0.4286,1,"{3777, 3922, 3920, 3774, 203, 3795}"
140,1,174,0,0.371567,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{35 \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{15 \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{35 \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{15 \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-5*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (115*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (15*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,21,0.3333,1,"{3817, 4020, 4022, 3920, 3774, 203, 3795}"
141,1,48,0,0.0380545,"\int \frac{\sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a - a*Sec[c + d*x]],x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d))","A",2,2,22,0.09091,1,"{3795, 203}"
142,1,87,0,0.0705244,"\int \frac{1}{\sqrt{a-a \sec (c+d x)}} \, dx","Int[1/Sqrt[a - a*Sec[c + d*x]],x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"(2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,4,15,0.2667,1,"{3776, 3774, 203, 3795}"
143,1,383,0,0.6800686,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(2/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 d}+\frac{57 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{80 d (\sec (c+d x)+1)}-\frac{19\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 d}+\frac{57 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{80 d (\sec (c+d x)+1)}-\frac{19\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(-9*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*d) + (57*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(80*d*(1 + Sec[c + d*x])) + (3*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*a*d) - (19*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(80*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",7,7,23,0.3043,1,"{3800, 4001, 3828, 3827, 50, 63, 225}"
144,1,353,0,0.352169,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(2/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d*(1 + Sec[c + d*x])) - (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",6,6,23,0.2609,1,"{3798, 3828, 3827, 50, 63, 225}"
145,1,326,0,0.2692339,"\int \sec (c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(2/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{2 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{2 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])) - (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",5,5,21,0.2381,1,"{3828, 3827, 50, 63, 225}"
146,1,77,0,0.0472338,"\int (a+a \sec (c+d x))^{2/3} \, dx","Int[(a + a*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}","\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}",1,"(3*Sqrt[2]*AppellF1[7/6, 1/2, 1, 13/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]])","A",3,3,14,0.2143,1,"{3779, 3778, 136}"
147,1,77,0,0.1080801,"\int \cos (c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(2/3),x]","-\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}","-\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}",1,"(-3*Sqrt[2]*AppellF1[7/6, 1/2, 2, 13/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]])","A",3,3,21,0.1429,1,"{3828, 3827, 136}"
148,1,413,0,0.5077378,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{8/3}}{11 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{88 d}+\frac{147 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{440 d}+\frac{1029 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{880 d (\sec (c+d x)+1)}-\frac{343\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{880 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{8/3}}{11 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{88 d}+\frac{147 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{440 d}+\frac{1029 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{880 d (\sec (c+d x)+1)}-\frac{343\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{880 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(147*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(440*d) + (1029*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(880*d*(1 + Sec[c + d*x])) - (9*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(88*d) + (3*(a + a*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*a*d) - (343*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(880*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",8,7,23,0.3043,1,"{3800, 4001, 3828, 3827, 50, 63, 225}"
149,1,383,0,0.3842969,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{8 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{16 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{16 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{8 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{16 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{16 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(3*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*d) + (21*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(16*d*(1 + Sec[c + d*x])) + (3*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*d) - (7*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(16*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",7,6,23,0.2609,1,"{3798, 3828, 3827, 50, 63, 225}"
150,1,356,0,0.2871369,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/3),x]","\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}","\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(3*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (21*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(10*d*(1 + Sec[c + d*x])) - (7*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",6,5,21,0.2381,1,"{3828, 3827, 50, 63, 225}"
151,1,86,0,0.0471119,"\int (a+a \sec (c+d x))^{5/3} \, dx","Int[(a + a*Sec[c + d*x])^(5/3),x]","\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},1;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}","\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},1;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}",1,"(3*Sqrt[2]*a*AppellF1[13/6, 1/2, 1, 19/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(13*d*Sqrt[1 - Sec[c + d*x]])","A",3,3,14,0.2143,1,"{3779, 3778, 136}"
152,1,86,0,0.1065846,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/3),x]","-\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},2;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}","-\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},2;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}",1,"(-3*Sqrt[2]*a*AppellF1[13/6, 1/2, 2, 19/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(13*d*Sqrt[1 - Sec[c + d*x]])","A",3,3,21,0.1429,1,"{3828, 3827, 136}"
153,1,371,0,0.5479114,"\int \frac{\sec ^4(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(1/3),x]","\frac{3 \tan (c+d x) \sec ^2(c+d x)}{8 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 a d}+\frac{99 \tan (c+d x)}{80 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{37\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}","\frac{3 \tan (c+d x) \sec ^2(c+d x)}{8 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 a d}+\frac{99 \tan (c+d x)}{80 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{37\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(99*Tan[c + d*x])/(80*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sec[c + d*x]^2*Tan[c + d*x])/(8*d*(a + a*Sec[c + d*x])^(1/3)) - (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*a*d) + (37*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(80*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",7,7,23,0.3043,1,"{3824, 4010, 4001, 3828, 3827, 63, 225}"
154,1,336,0,0.394185,"\int \frac{\sec ^3(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(1/3),x]","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}-\frac{9 \tan (c+d x)}{10 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{7\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}-\frac{9 \tan (c+d x)}{10 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{7\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(-9*Tan[c + d*x])/(10*d*(a + a*Sec[c + d*x])^(1/3)) + (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*a*d) - (7*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",6,6,23,0.2609,1,"{3800, 4001, 3828, 3827, 63, 225}"
155,1,306,0,0.30231,"\int \frac{\sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(1/3),x]","\frac{3 \tan (c+d x)}{2 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}","\frac{3 \tan (c+d x)}{2 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",5,5,23,0.2174,1,"{3798, 3828, 3827, 63, 225}"
156,1,276,0,0.2279503,"\int \frac{\sec (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^(1/3),x]","-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}","-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"-((3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]))","A",4,4,21,0.1905,1,"{3828, 3827, 63, 225}"
157,1,75,0,0.042687,"\int \frac{1}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(-1/3),x]","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}",1,"(3*Sqrt[2]*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3))","A",3,3,14,0.2143,1,"{3779, 3778, 136}"
158,1,75,0,0.0943261,"\int \frac{\cos (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^(1/3),x]","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},2;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},2;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}",1,"(-3*Sqrt[2]*AppellF1[1/6, 1/2, 2, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3))","A",3,3,21,0.1429,1,"{3828, 3827, 136}"
159,1,766,0,1.0586193,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/3),x]","\frac{375 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}-\frac{125\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{375 \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac{33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}}","\frac{375 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}-\frac{125\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{375 \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac{33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}}",1,"(-33*Tan[c + d*x])/(28*d*(a + a*Sec[c + d*x])^(5/3)) + (3*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/3)) + (135*Tan[c + d*x])/(14*a*d*(a + a*Sec[c + d*x])^(2/3)) + (375*(1 + Sqrt[3])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*a^2*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (375*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(14*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (125*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(28*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,23,0.3913,1,"{3824, 4008, 4000, 3828, 3827, 63, 308, 225, 1881}"
160,1,731,0,0.770587,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/3),x]","-\frac{57 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{7 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{19\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{57 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{36 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}","-\frac{57 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{7 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{19\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{57 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{36 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}",1,"(3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) - (36*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (57*(1 + Sqrt[3])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a^2*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (57*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (19*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",8,8,23,0.3478,1,"{3799, 4000, 3828, 3827, 63, 308, 225, 1881}"
161,1,731,0,0.6737416,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/3),x]","\frac{15 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{15 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}-\frac{5\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{15 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}","\frac{15 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{15 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}-\frac{5\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{15 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}",1,"(-3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) + (15*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) + (15*(1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (15*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (5*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",8,8,23,0.3478,1,"{3797, 3828, 3827, 51, 63, 308, 225, 1881}"
162,1,744,0,0.5798125,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]/(a + a*Sec[c + d*x])^(5/3),x]","\frac{6 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 a d (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}+\frac{6 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{\sqrt[3]{2} 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{6 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}","\frac{6 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 a d (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}+\frac{6 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{\sqrt[3]{2} 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{6 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}",1,"(6*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) + (3*Tan[c + d*x])/(7*a*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)) + (6*(1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (6*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (2^(1/3)*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",8,7,21,0.3333,1,"{3828, 3827, 51, 63, 308, 225, 1881}"
163,1,90,0,0.0476895,"\int \frac{1}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[(a + a*Sec[c + d*x])^(-5/3),x]","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},1;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},1;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}",1,"(-3*Sqrt[2]*AppellF1[-7/6, 1/2, 1, -1/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(7*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3))","A",3,3,14,0.2143,1,"{3779, 3778, 136}"
164,1,90,0,0.1108469,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[Cos[c + d*x]/(a + a*Sec[c + d*x])^(5/3),x]","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},2;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},2;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}",1,"(3*Sqrt[2]*AppellF1[-7/6, 1/2, 2, -1/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(7*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3))","A",3,3,21,0.1429,1,"{3828, 3827, 136}"
165,1,151,0,0.0905414,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]),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",8,5,21,0.2381,1,"{3787, 3768, 3771, 2641, 2639}"
166,1,123,0,0.0763306,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]),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",7,5,21,0.2381,1,"{3787, 3768, 3771, 2639, 2641}"
167,1,97,0,0.0644187,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]),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",6,5,21,0.2381,1,"{3787, 3771, 2641, 3768, 2639}"
168,1,75,0,0.0574911,"\int \frac{a+a \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[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",5,4,21,0.1905,1,"{3787, 3771, 2639, 2641}"
169,1,101,0,0.0692706,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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 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",6,5,21,0.2381,1,"{3787, 3769, 3771, 2641, 2639}"
170,1,127,0,0.082885,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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)}{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",7,5,21,0.2381,1,"{3787, 3769, 3771, 2639, 2641}"
171,1,151,0,0.0953708,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])/Sec[c + d*x]^(7/2),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",8,5,21,0.2381,1,"{3787, 3769, 3771, 2641, 2639}"
172,1,187,0,0.1317821,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2,x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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)}{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{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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)}{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) + (12*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(7*d) + (4*a^2*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",9,6,23,0.2609,1,"{3788, 3768, 3771, 2639, 4046, 2641}"
173,1,161,0,0.1144708,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^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",8,6,23,0.2609,1,"{3788, 3768, 3771, 2641, 4046, 2639}"
174,1,131,0,0.101867,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^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",7,6,23,0.2609,1,"{3788, 3768, 3771, 2639, 4046, 2641}"
175,1,64,0,0.0736912,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^2/Sqrt[Sec[c + d*x]],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",4,4,23,0.1739,1,"{3788, 3771, 2641, 4043}"
176,1,107,0,0.0917851,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(3/2),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",6,5,23,0.2174,1,"{3788, 3771, 2639, 4045, 2641}"
177,1,135,0,0.1054211,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(5/2),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",7,6,23,0.2609,1,"{3788, 3769, 3771, 2641, 4045, 2639}"
178,1,161,0,0.114728,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(7/2),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",8,6,23,0.2609,1,"{3788, 3769, 3771, 2639, 4045, 2641}"
179,1,187,0,0.1898809,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3,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",16,5,23,0.2174,1,"{3791, 3768, 3771, 2639, 2641}"
180,1,157,0,0.1609605,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3,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",14,5,23,0.2174,1,"{3791, 3771, 2641, 3768, 2639}"
181,1,131,0,0.1416459,"\int \frac{(a+a \sec (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^3/Sqrt[Sec[c + d*x]],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",12,5,23,0.2174,1,"{3791, 3771, 2639, 2641, 3768}"
182,1,131,0,0.1378185,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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",12,6,23,0.2609,1,"{3791, 3769, 3771, 2641, 2639, 3768}"
183,1,131,0,0.1395512,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(5/2),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",12,5,23,0.2174,1,"{3791, 3769, 3771, 2639, 2641}"
184,1,161,0,0.1676629,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(7/2),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",14,5,23,0.2174,1,"{3791, 3769, 3771, 2641, 2639}"
185,1,187,0,0.1943081,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(9/2),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",16,5,23,0.2174,1,"{3791, 3769, 3771, 2639, 2641}"
186,1,213,0,0.2514935,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4,x]","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{152 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 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)}{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{2 a^4 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{152 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 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)}{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) + (152*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (32*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(7*d) + (122*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (8*a^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + (2*a^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",21,5,23,0.2174,1,"{3791, 3768, 3771, 2639, 2641}"
187,1,187,0,0.2094005,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^4 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^4,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",18,5,23,0.2174,1,"{3791, 3771, 2641, 3768, 2639}"
188,1,161,0,0.181752,"\int \frac{(a+a \sec (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^4/Sqrt[Sec[c + d*x]],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",16,5,23,0.2174,1,"{3791, 3771, 2639, 2641, 3768}"
189,1,118,0,0.1693673,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(3/2),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",15,6,23,0.2609,1,"{3791, 3769, 3771, 2641, 2639, 3768}"
190,1,159,0,0.1718758,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(5/2),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",15,6,23,0.2609,1,"{3791, 3769, 3771, 2639, 2641, 3768}"
191,1,161,0,0.1870037,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(7/2),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",16,5,23,0.2174,1,"{3791, 3769, 3771, 2641, 2639}"
192,1,187,0,0.2219836,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(9/2),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",18,5,23,0.2174,1,"{3791, 3769, 3771, 2639, 2641}"
193,1,213,0,0.2582382,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]","\frac{128 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{150 a^4 \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{904 a^4 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{904 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 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{128 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{150 a^4 \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{904 a^4 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{904 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 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,"(128*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (904*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (8*a^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (150*a^4*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (128*a^4*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (904*a^4*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",21,5,23,0.2174,1,"{3791, 3769, 3771, 2641, 2639}"
194,1,164,0,0.1264842,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(7/2)/(a + a*Sec[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",8,6,23,0.2609,1,"{3818, 3787, 3768, 3771, 2639, 2641}"
195,1,136,0,0.1125247,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Sec[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",7,6,23,0.2609,1,"{3818, 3787, 3771, 2641, 3768, 2639}"
196,1,110,0,0.1020414,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*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",6,5,23,0.2174,1,"{3818, 3787, 3771, 2639, 2641}"
197,1,110,0,0.1000471,"\int \frac{\sqrt{\sec (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*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",6,5,23,0.2174,1,"{3820, 3787, 3771, 2639, 2641}"
198,1,112,0,0.1018856,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + a*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{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",6,5,23,0.2174,1,"{3819, 3787, 3771, 2639, 2641}"
199,1,140,0,0.1168448,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),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",7,6,23,0.2609,1,"{3819, 3787, 3769, 3771, 2641, 2639}"
200,1,168,0,0.1294674,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),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",8,6,23,0.2609,1,"{3819, 3787, 3769, 3771, 2639, 2641}"
201,1,202,0,0.2291491,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(9/2)/(a + a*Sec[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",9,7,23,0.3043,1,"{3816, 4019, 3787, 3768, 3771, 2639, 2641}"
202,1,176,0,0.2135484,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(7/2)/(a + a*Sec[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",8,7,23,0.3043,1,"{3816, 4019, 3787, 3771, 2641, 3768, 2639}"
203,1,149,0,0.1986493,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Sec[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",7,6,23,0.2609,1,"{3816, 4019, 3787, 3771, 2639, 2641}"
204,1,77,0,0.0609247,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^2,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",4,4,23,0.1739,1,"{3815, 21, 3771, 2641}"
205,1,149,0,0.2011465,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Sec[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",7,6,23,0.2609,1,"{3817, 4019, 3787, 3771, 2639, 2641}"
206,1,152,0,0.2021537,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^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",7,6,23,0.2609,1,"{3817, 4020, 3787, 3771, 2639, 2641}"
207,1,178,0,0.2247893,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^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",8,7,23,0.3043,1,"{3817, 4020, 3787, 3769, 3771, 2641, 2639}"
208,1,200,0,0.2397561,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^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",9,7,23,0.3043,1,"{3817, 4020, 3787, 3769, 3771, 2639, 2641}"
209,1,247,0,0.35645,"\int \frac{\sec ^{\frac{11}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(11/2)/(a + a*Sec[c + d*x])^3,x]","-\frac{119 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\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{\sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}","-\frac{119 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\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{\sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}",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) - (119*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + (11*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - (Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))","A",10,7,23,0.3043,1,"{3816, 4019, 3787, 3768, 3771, 2639, 2641}"
210,1,221,0,0.3395402,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(9/2)/(a + a*Sec[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",9,7,23,0.3043,1,"{3816, 4019, 3787, 3771, 2641, 3768, 2639}"
211,1,195,0,0.3246766,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(7/2)/(a + a*Sec[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",8,6,23,0.2609,1,"{3816, 4019, 3787, 3771, 2639, 2641}"
212,1,195,0,0.3200136,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^3,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",8,7,23,0.3043,1,"{3816, 4019, 4020, 3787, 3771, 2639, 2641}"
213,1,195,0,0.3194627,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^3,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",8,7,23,0.3043,1,"{3815, 4019, 4020, 3787, 3771, 2639, 2641}"
214,1,195,0,0.3290343,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^3} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^3,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",8,7,23,0.3043,1,"{3817, 4019, 4020, 3787, 3771, 2639, 2641}"
215,1,195,0,0.3218507,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),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",8,6,23,0.2609,1,"{3817, 4020, 3787, 3771, 2639, 2641}"
216,1,221,0,0.3517902,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),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",9,7,23,0.3043,1,"{3817, 4020, 3787, 3769, 3771, 2641, 2639}"
217,1,247,0,0.3737654,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{63 \sin (c+d x)}{10 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}+\frac{77 \sin (c+d x)}{10 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{21 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{21 \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{231 \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{4 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}","-\frac{63 \sin (c+d x)}{10 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}+\frac{77 \sin (c+d x)}{10 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{21 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{21 \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{231 \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{4 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(231*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (21*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (77*Sin[c + d*x])/(10*a^3*d*Sec[c + d*x]^(3/2)) - (21*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (4*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (63*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))","A",10,7,23,0.3043,1,"{3817, 4020, 3787, 3769, 3771, 2639, 2641}"
218,1,116,0,0.1672407,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{3 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{3 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(3*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (3*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",4,3,25,0.1200,1,"{3803, 3801, 215}"
219,1,72,0,0.112704,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3803, 3801, 215}"
220,1,37,0,0.0583492,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d","A",2,2,25,0.08000,1,"{3801, 215}"
221,1,36,0,0.0541663,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",1,1,25,0.04000,1,"{3804}"
222,1,77,0,0.1089705,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3805, 3804}"
223,1,115,0,0.1637376,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Sec[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) \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",3,2,25,0.08000,1,"{3805, 3804}"
224,1,153,0,0.2156771,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]","\frac{12 a \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{35 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{12 a \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{35 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (12*a*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (32*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]])","A",4,2,25,0.08000,1,"{3805, 3804}"
225,1,160,0,0.2343746,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}","\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}",1,"(11*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (11*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,25,0.2000,1,"{3814, 21, 3803, 3801, 215}"
226,1,120,0,0.1751603,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(7*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (7*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3814, 21, 3803, 3801, 215}"
227,1,75,0,0.1174627,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(3*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3814, 21, 3801, 215}"
228,1,76,0,0.11753,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3813, 21, 3801, 215}"
229,1,79,0,0.1089232,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3809, 3804}"
230,1,116,0,0.1721583,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2),x]","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sqrt{\sec (c+d x)}}",1,"(8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",3,3,25,0.1200,1,"{3812, 3809, 3804}"
231,1,161,0,0.2323063,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]","\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (26*a^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (104*a^2*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (208*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,25,0.1600,1,"{3813, 21, 3805, 3804}"
232,1,201,0,0.2917533,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2),x]","\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{544 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{544 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (34*a^2*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (68*a^2*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (272*a^2*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (544*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]])","A",6,4,25,0.1600,1,"{3813, 21, 3805, 3804}"
233,1,200,0,0.3370868,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{17 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}","\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{17 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}",1,"(163*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (163*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (17*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,25,0.2000,1,"{3814, 4016, 3803, 3801, 215}"
234,1,160,0,0.2749746,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{13 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{25 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{25 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}","\frac{13 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{25 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{25 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}",1,"(25*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (25*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (13*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{3814, 4016, 3803, 3801, 215}"
235,1,120,0,0.2150839,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{9 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{19 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{9 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{19 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(19*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (9*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,25,0.1600,1,"{3814, 4016, 3801, 215}"
236,1,112,0,0.2169748,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}+\frac{5 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}+\frac{5 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(5*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,25,0.1600,1,"{3814, 4015, 3801, 215}"
237,1,118,0,0.2190594,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (14*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3813, 4015, 3801, 215}"
238,1,119,0,0.1694188,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2),x]","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",3,2,25,0.08000,1,"{3809, 3804}"
239,1,156,0,0.2361411,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2),x]","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",4,3,25,0.1200,1,"{3812, 3809, 3804}"
240,1,201,0,0.3327425,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]","\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(38*a^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (146*a^3*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (584*a^3*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (1168*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,4,25,0.1600,1,"{3813, 4015, 3805, 3804}"
241,1,241,0,0.4032434,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(11/2),x]","\frac{284 a^3 \sin (c+d x)}{231 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{46 a^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2272 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{1136 a^3 \sin (c+d x)}{693 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{284 a^3 \sin (c+d x)}{231 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{46 a^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2272 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{1136 a^3 \sin (c+d x)}{693 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(46*a^3*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (710*a^3*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (284*a^3*Sin[c + d*x])/(231*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (1136*a^3*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2272*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,4,25,0.1600,1,"{3813, 4015, 3805, 3804}"
242,1,38,0,0.0564871,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt[4]{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/4),x]","\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}","\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(4*a^2*Sec[c + d*x]^(3/4)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3814, 8}"
243,1,37,0,0.0576645,"\int \sqrt{\sec (e+f x)} \sqrt{a+a \sec (e+f x)} \, dx","Int[Sqrt[Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}",1,"(2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f","A",2,2,25,0.08000,1,"{3801, 215}"
244,1,38,0,0.0633156,"\int \sqrt{-\sec (e+f x)} \sqrt{a-a \sec (e+f x)} \, dx","Int[Sqrt[-Sec[e + f*x]]*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right)}{f}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right)}{f}",1,"(2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a - a*Sec[e + f*x]]])/f","A",2,2,28,0.07143,1,"{3801, 215}"
245,1,128,0,0.2666605,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-(ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{3822, 4023, 3808, 206, 3801, 215}"
246,1,95,0,0.1638479,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",5,5,25,0.2000,1,"{3821, 3801, 215, 3808, 206}"
247,1,56,0,0.0594848,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",2,2,25,0.08000,1,"{3808, 206}"
248,1,93,0,0.1118313,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3812, 3808, 206}"
249,1,131,0,0.21943,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3823, 4013, 3808, 206}"
250,1,169,0,0.3449496,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3823, 4022, 4013, 3808, 206}"
251,1,174,0,0.4192759,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}","\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}",1,"(-3*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (9*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3816, 4021, 4023, 3808, 206, 3801, 215}"
252,1,134,0,0.2829675,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,6,25,0.2400,1,"{3816, 4023, 3808, 206, 3801, 215}"
253,1,97,0,0.1228964,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,25,0.1200,1,"{3810, 3808, 206}"
254,1,97,0,0.1219894,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",3,3,25,0.1200,1,"{3811, 3808, 206}"
255,1,137,0,0.2347739,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}","-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(-7*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{3817, 4013, 3808, 206}"
256,1,177,0,0.3669355,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{7 \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}","\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{7 \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(11*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (7*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3817, 4022, 4013, 3808, 206}"
257,1,217,0,0.515508,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{13 \sin (c+d x)}{10 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{13 \sin (c+d x)}{10 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-15*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + (9*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (13*Sin[c + d*x])/(10*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,25,0.2000,1,"{3817, 4022, 4013, 3808, 206}"
258,1,214,0,0.5620648,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{35 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","\frac{35 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-5*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (115*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (15*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (35*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3816, 4019, 4021, 4023, 3808, 206, 3801, 215}"
259,1,174,0,0.4298072,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","-\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (11*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,7,25,0.2800,1,"{3816, 4019, 4023, 3808, 206, 3801, 215}"
260,1,137,0,0.1868494,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,3,25,0.1200,1,"{3810, 3808, 206}"
261,1,137,0,0.1881266,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{3811, 3810, 3808, 206}"
262,1,137,0,0.2421292,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(19*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (9*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{3817, 4012, 3808, 206}"
263,1,177,0,0.37984,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-75*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3817, 4020, 4013, 3808, 206}"
264,1,217,0,0.5071704,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}","-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(163*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - (17*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (95*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (299*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{3817, 4020, 4022, 4013, 3808, 206}"
265,1,126,0,0.2853094,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(7/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{\sec (c+d x)+1}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}+\frac{7 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{4 d}","\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{\sec (c+d x)+1}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}+\frac{7 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{4 d}",1,"-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (7*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/(4*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Sec[c + d*x]])","A",7,6,23,0.2609,1,"{3822, 4021, 4023, 3807, 215, 3801}"
266,1,85,0,0.1853321,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}-\frac{\sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}-\frac{\sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}",1,"(Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d - ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]]/d + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]])","A",6,5,23,0.2174,1,"{3822, 4023, 3807, 215, 3801}"
267,1,54,0,0.1094096,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{2 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}","\frac{2 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/d","A",5,4,23,0.1739,1,"{3821, 3801, 215, 3807}"
268,1,27,0,0.0391462,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[1 + Sec[c + d*x]],x]","\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}","\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"(Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d","A",2,2,23,0.08696,1,"{3807, 215}"
269,1,62,0,0.078271,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]),x]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]])","A",3,3,23,0.1304,1,"{3812, 3807, 215}"
270,1,98,0,0.1514968,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{1+\sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]]),x]","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\sec (c+d x)+1}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\sec (c+d x)+1}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"(Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[1 + Sec[c + d*x]])","A",4,4,23,0.1739,1,"{3823, 4013, 3807, 215}"
271,1,134,0,0.234701,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{1+\sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*Sqrt[1 + Sec[c + d*x]]),x]","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)+1}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\sec (c+d x)+1}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)+1}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\sec (c+d x)+1}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[1 + Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3823, 4022, 4013, 3807, 215}"
272,1,325,0,0.3833541,"\int (e \sec (c+d x))^{4/3} \sqrt{a+a \sec (c+d x)} \, dx","Int[(e*Sec[c + d*x])^(4/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 e \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{5 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{6 a e \tan (c+d x) \sqrt[3]{e \sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}","\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 e \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{5 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{6 a e \tan (c+d x) \sqrt[3]{e \sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(6*a*e*(e*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*e*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",4,4,27,0.1481,1,"{3806, 50, 63, 218}"
273,1,280,0,0.1980756,"\int \sqrt[3]{e \sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Int[(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}",1,"(2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",3,3,27,0.1111,1,"{3806, 63, 218}"
274,1,326,0,0.2436189,"\int \frac{\sqrt{a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(2/3),x]","\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{2 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{2 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}",1,"(3*a*Tan[c + d*x])/(2*d*(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]]) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*d*e*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",4,4,27,0.1481,1,"{3806, 51, 63, 218}"
275,1,716,0,0.560383,"\int (e \sec (c+d x))^{8/3} \sqrt{a+a \sec (c+d x)} \, dx","Int[(e*Sec[c + d*x])^(8/3)*Sqrt[a + a*Sec[c + d*x]],x]","-\frac{80 \sqrt{2} 3^{3/4} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{120 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{240 a e^3 \tan (c+d x)}{91 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{60 a e^2 \tan (c+d x) (e \sec (c+d x))^{2/3}}{91 d \sqrt{a \sec (c+d x)+a}}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{5/3}}{13 d \sqrt{a \sec (c+d x)+a}}","-\frac{80 \sqrt{2} 3^{3/4} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{120 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{240 a e^3 \tan (c+d x)}{91 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{60 a e^2 \tan (c+d x) (e \sec (c+d x))^{2/3}}{91 d \sqrt{a \sec (c+d x)+a}}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{5/3}}{13 d \sqrt{a \sec (c+d x)+a}}",1,"(60*a*e^2*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(91*d*Sqrt[a + a*Sec[c + d*x]]) + (6*a*e*(e*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(13*d*Sqrt[a + a*Sec[c + d*x]]) - (240*a*e^3*Tan[c + d*x])/(91*d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) + (120*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(7/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(91*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (80*Sqrt[2]*3^(3/4)*a^2*e^(7/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(91*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",7,6,27,0.2222,1,"{3806, 50, 63, 303, 218, 1877}"
276,1,673,0,0.4783002,"\int (e \sec (c+d x))^{5/3} \sqrt{a+a \sec (c+d x)} \, dx","Int[(e*Sec[c + d*x])^(5/3)*Sqrt[a + a*Sec[c + d*x]],x]","-\frac{8 \sqrt{2} 3^{3/4} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{24 a e^2 \tan (c+d x)}{7 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{2/3}}{7 d \sqrt{a \sec (c+d x)+a}}","-\frac{8 \sqrt{2} 3^{3/4} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{24 a e^2 \tan (c+d x)}{7 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{2/3}}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(6*a*e*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (24*a*e^2*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) + (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(4/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (8*Sqrt[2]*3^(3/4)*a^2*e^(4/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",6,6,27,0.2222,1,"{3806, 50, 63, 303, 218, 1877}"
277,1,624,0,0.4352545,"\int (e \sec (c+d x))^{2/3} \sqrt{a+a \sec (c+d x)} \, dx","Int[(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 \sqrt{2} 3^{3/4} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{6 a e \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}","-\frac{2 \sqrt{2} 3^{3/4} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{6 a e \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(-6*a*e*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(1/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (2*Sqrt[2]*3^(3/4)*a^2*e^(1/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",5,5,27,0.1852,1,"{3806, 63, 303, 218, 1877}"
278,1,662,0,0.4833586,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt[3]{e \sec (c+d x)}} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(1/3),x]","\frac{\sqrt{2} 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}","\frac{\sqrt{2} 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(3*a*Tan[c + d*x])/(d*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*d*e^(2/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) + (Sqrt[2]*3^(3/4)*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*e^(2/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",6,6,27,0.2222,1,"{3806, 51, 63, 303, 218, 1877}"
279,1,715,0,0.5398896,"\int \frac{\sqrt{a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(4/3),x]","\frac{5\ 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{16 d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{4 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{4/3}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}","\frac{5\ 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{16 d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{4 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{4/3}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(3*a*Tan[c + d*x])/(4*d*(e*Sec[c + d*x])^(4/3)*Sqrt[a + a*Sec[c + d*x]]) + (15*a*Tan[c + d*x])/(8*d*e*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]) + (15*a*Tan[c + d*x])/(8*d*e*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) - (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(16*d*e^(5/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) + (5*3^(3/4)*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*Sqrt[2]*d*e^(5/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])","A",7,6,27,0.2222,1,"{3806, 51, 63, 303, 218, 1877}"
280,1,78,0,0.1643339,"\int \frac{(e \sec (c+d x))^{2/3}}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(e*Sec[c + d*x])^(2/3)/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{3 \tan (c+d x) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right) (e \sec (c+d x))^{2/3}}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","-\frac{3 \tan (c+d x) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right) (e \sec (c+d x))^{2/3}}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-3*AppellF1[2/3, 1/2, 1, 5/3, Sec[c + d*x], -Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",4,4,27,0.1481,1,"{3828, 3827, 130, 510}"
281,1,76,0,0.1421689,"\int \frac{\sqrt[3]{e \sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(e*Sec[c + d*x])^(1/3)/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{3 \tan (c+d x) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\sec (c+d x),-\sec (c+d x)\right) \sqrt[3]{e \sec (c+d x)}}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}","-\frac{3 \tan (c+d x) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\sec (c+d x),-\sec (c+d x)\right) \sqrt[3]{e \sec (c+d x)}}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-3*AppellF1[1/3, 1/2, 1, 4/3, Sec[c + d*x], -Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",4,4,27,0.1481,1,"{3828, 3827, 130, 429}"
282,1,76,0,0.1589702,"\int \frac{1}{\sqrt[3]{e \sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/((e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{3 \tan (c+d x) F_1\left(-\frac{1}{3};\frac{1}{2},1;\frac{2}{3};\sec (c+d x),-\sec (c+d x)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}","\frac{3 \tan (c+d x) F_1\left(-\frac{1}{3};\frac{1}{2},1;\frac{2}{3};\sec (c+d x),-\sec (c+d x)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}",1,"(3*AppellF1[-1/3, 1/2, 1, 2/3, Sec[c + d*x], -Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]])","A",4,4,27,0.1481,1,"{3828, 3827, 130, 510}"
283,1,78,0,0.168797,"\int \frac{1}{(e \sec (c+d x))^{2/3} \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/((e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{3 \tan (c+d x) F_1\left(-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};\sec (c+d x),-\sec (c+d x)\right)}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}","\frac{3 \tan (c+d x) F_1\left(-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};\sec (c+d x),-\sec (c+d x)\right)}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}",1,"(3*AppellF1[-2/3, 1/2, 1, 1/3, Sec[c + d*x], -Sec[c + d*x]]*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]])","A",4,4,27,0.1481,1,"{3828, 3827, 130, 510}"
284,1,78,0,0.1094527,"\int \sec ^{\frac{4}{3}}(c+d x) \sqrt[3]{a+a \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(1/3),x]","\frac{2^{5/6} \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};-\frac{1}{3},\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}","\frac{2^{5/6} \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};-\frac{1}{3},\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}",1,"(2^(5/6)*AppellF1[1/2, -1/3, 1/6, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(5/6))","A",3,3,25,0.1200,1,"{3828, 3825, 133}"
285,1,79,0,0.1238679,"\int \sec ^{\frac{4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(2/3),x]","\frac{2 \sqrt[6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{1}{3},-\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{7/6}}","\frac{2 \sqrt[6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{1}{3},-\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{7/6}}",1,"(2*2^(1/6)*AppellF1[1/2, -1/3, -1/6, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(7/6))","A",3,3,25,0.1200,1,"{3828, 3825, 133}"
286,1,79,0,0.1220505,"\int \sec ^{\frac{5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^(5/3)*(a + a*Sec[c + d*x])^(2/3),x]","\frac{2 \sqrt[6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{2}{3},-\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{7/6}}","-\frac{5 \tan ^3(c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}{8 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{10/3}}+\frac{\tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{5}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}{8 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{4/3}}+\frac{9 \sin (c+d x) \sec ^{\frac{2}{3}}(c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d}-\frac{3 a \sin (c+d x) \sec ^{\frac{5}{3}}(c+d x)}{2 d \sqrt[3]{a (\sec (c+d x)+1)}}-\frac{9 \tan (c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{7/3}}",1,"(2*2^(1/6)*AppellF1[1/2, -2/3, -1/6, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(7/6))","C",3,3,25,0.1200,0,"{3828, 3825, 133}"
287,1,80,0,0.13262,"\int \frac{(a+a \sec (c+d x))^{4/3}}{\sqrt[3]{\sec (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(4/3)/Sec[c + d*x]^(1/3),x]","\frac{2\ 2^{5/6} a \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};\frac{4}{3},-\frac{5}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}","\frac{2\ 2^{5/6} a \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};\frac{4}{3},-\frac{5}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}",1,"(2*2^(5/6)*a*AppellF1[1/2, 4/3, -5/6, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(5/6))","A",3,3,25,0.1200,1,"{3828, 3825, 133}"
288,1,304,0,0.4846462,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^4 \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^4,x]","-\frac{a^4 \left(8 n^2+24 n+3\right) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) (n+3) \sqrt{\sin ^2(e+f x)}}+\frac{4 a^4 (2 n+3) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^4 \left(4 n^2+21 n+30\right) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2) (n+3)}+\frac{\sin (e+f x) \left(a^2 \sec (e+f x)+a^2\right)^2 \sec ^{n+1}(e+f x)}{f (n+3)}+\frac{2 (n+4) \sin (e+f x) \left(a^4 \sec (e+f x)+a^4\right) \sec ^{n+1}(e+f x)}{f (n+2) (n+3)}","-\frac{a^4 \left(8 n^2+24 n+3\right) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) (n+3) \sqrt{\sin ^2(e+f x)}}+\frac{4 a^4 (2 n+3) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^4 \left(4 n^2+21 n+30\right) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2) (n+3)}+\frac{\sin (e+f x) \left(a^2 \sec (e+f x)+a^2\right)^2 \sec ^{n+1}(e+f x)}{f (n+3)}+\frac{2 (n+4) \sin (e+f x) \left(a^4 \sec (e+f x)+a^4\right) \sec ^{n+1}(e+f x)}{f (n+2) (n+3)}",1,"(a^4*(30 + 21*n + 4*n^2)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)*(2 + n)*(3 + n)) + (Sec[e + f*x]^(1 + n)*(a^2 + a^2*Sec[e + f*x])^2*Sin[e + f*x])/(f*(3 + n)) + (2*(4 + n)*Sec[e + f*x]^(1 + n)*(a^4 + a^4*Sec[e + f*x])*Sin[e + f*x])/(f*(2 + n)*(3 + n)) - (a^4*(3 + 24*n + 8*n^2)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*(1 + n)*(3 + n)*Sqrt[Sin[e + f*x]^2]) + (4*a^4*(3 + 2*n)*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2])","A",8,6,21,0.2857,1,"{3814, 4018, 3997, 3787, 3772, 2643}"
289,1,230,0,0.2761062,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^3,x]","-\frac{a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac{\sin (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \sec ^{n+1}(e+f x)}{f (n+2)}","-\frac{a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac{\sin (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \sec ^{n+1}(e+f x)}{f (n+2)}",1,"(a^3*(5 + 2*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)*(2 + n)) + (Sec[e + f*x]^(1 + n)*(a^3 + a^3*Sec[e + f*x])*Sin[e + f*x])/(f*(2 + n)) - (a^3*(1 + 4*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(7 + 4*n)*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2])","A",7,5,21,0.2381,1,"{3814, 3997, 3787, 3772, 2643}"
290,1,172,0,0.1396752,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^2 \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^2,x]","-\frac{a^2 (2 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1)}","-\frac{a^2 (2 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1)}",1,"(a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)) - (a^2*(1 + 2*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (2*a^2*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])","A",6,4,21,0.1905,1,"{3788, 3772, 2643, 4046}"
291,1,132,0,0.0865364,"\int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x]),x]","\frac{a \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}","\frac{a \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"-((a*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (a*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])","A",5,3,19,0.1579,1,"{3787, 3772, 2643}"
292,1,174,0,0.1575347,"\int \frac{\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx","Int[Sec[e + f*x]^n/(a + a*Sec[e + f*x]),x]","\frac{(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(e+f x)\right)}{a f (2-n) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)}","\frac{(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(e+f x)\right)}{a f (2-n) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)}",1,"(Sec[e + f*x]^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) + ((1 - n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-2 + n)*Sin[e + f*x])/(a*f*(2 - n)*Sqrt[Sin[e + f*x]^2]) - (Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2])","A",6,4,21,0.1905,1,"{3820, 3787, 3772, 2643}"
293,1,217,0,0.3089239,"\int \frac{\sec ^n(e+f x)}{(a+a \sec (e+f x))^2} \, dx","Int[Sec[e + f*x]^n/(a + a*Sec[e + f*x])^2,x]","-\frac{(3-2 n) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}+\frac{2 (2-n) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n) \sin (e+f x) \sec ^{n+1}(e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{\sin (e+f x) \sec ^{n+1}(e+f x)}{3 f (a \sec (e+f x)+a)^2}","-\frac{(3-2 n) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}+\frac{2 (2-n) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n) \sin (e+f x) \sec ^{n+1}(e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{\sin (e+f x) \sec ^{n+1}(e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"(-2*(2 - n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - (Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - ((3 - 2*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) + (2*(2 - n)*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2])","A",7,5,21,0.2381,1,"{3817, 4020, 3787, 3772, 2643}"
294,1,162,0,0.2419285,"\int \sec ^n(e+f x) (1+\sec (e+f x))^{5/2} \, dx","Int[Sec[e + f*x]^n*(1 + Sec[e + f*x])^(5/2),x]","\frac{2 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sqrt{\sec (e+f x)+1} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac{2 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}","\frac{2 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sqrt{\sec (e+f x)+1} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac{2 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}",1,"(2*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[1 + Sec[e + f*x]]) + (2*Sec[e + f*x]^(1 + n)*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[1 + Sec[e + f*x]])","A",4,4,21,0.1905,1,"{3814, 4016, 3806, 65}"
295,1,98,0,0.116606,"\int \sec ^n(e+f x) (1+\sec (e+f x))^{3/2} \, dx","Int[Sec[e + f*x]^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}","\frac{2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}",1,"(2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) + (2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]])","A",4,4,21,0.1905,1,"{3814, 21, 3806, 65}"
296,1,45,0,0.0517975,"\int \sec ^n(e+f x) \sqrt{1+\sec (e+f x)} \, dx","Int[Sec[e + f*x]^n*Sqrt[1 + Sec[e + f*x]],x]","\frac{2 \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}","\frac{2 \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",2,2,21,0.09524,1,"{3806, 65}"
297,1,59,0,0.0769604,"\int \frac{\sec ^n(e+f x)}{\sqrt{1+\sec (e+f x)}} \, dx","Int[Sec[e + f*x]^n/Sqrt[1 + Sec[e + f*x]],x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(AppellF1[1/2, 1 - n, 1, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",3,3,21,0.1429,1,"{3825, 130, 429}"
298,1,62,0,0.0802565,"\int \frac{\sec ^n(e+f x)}{(1+\sec (e+f x))^{3/2}} \, dx","Int[Sec[e + f*x]^n/(1 + Sec[e + f*x])^(3/2),x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 f \sqrt{\sec (e+f x)+1}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 f \sqrt{\sec (e+f x)+1}}",1,"(AppellF1[1/2, 1 - n, 2, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*Tan[e + f*x])/(2*f*Sqrt[1 + Sec[e + f*x]])","A",3,3,21,0.1429,1,"{3825, 130, 429}"
299,1,117,0,0.1294732,"\int (-\sec (e+f x))^n (1+\sec (e+f x))^{3/2} \, dx","Int[(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","\frac{2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) - ((1 + 4*n)*Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*(1 + 2*n)*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3814, 21, 3806, 64}"
300,1,64,0,0.0548234,"\int (-\sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx","Int[(-Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]","-\frac{\tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3806, 64}"
301,1,73,0,0.0693781,"\int \frac{(-\sec (e+f x))^n}{\sqrt{1+\sec (e+f x)}} \, dx","Int[(-Sec[e + f*x])^n/Sqrt[1 + Sec[e + f*x]],x]","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3826, 136}"
302,1,73,0,0.0736466,"\int \frac{(-\sec (e+f x))^n}{(1+\sec (e+f x))^{3/2}} \, dx","Int[(-Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2),x]","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3826, 136}"
303,1,117,0,0.1252071,"\int (d \sec (e+f x))^n (1+\sec (e+f x))^{3/2} \, dx","Int[(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","\frac{2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) - ((1 + 4*n)*Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*(1 + 2*n)*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3814, 21, 3806, 64}"
304,1,64,0,0.0566329,"\int (d \sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx","Int[(d*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]","-\frac{\tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3806, 64}"
305,1,73,0,0.0685992,"\int \frac{(d \sec (e+f x))^n}{\sqrt{1+\sec (e+f x)}} \, dx","Int[(d*Sec[e + f*x])^n/Sqrt[1 + Sec[e + f*x]],x]","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3827, 133}"
306,1,73,0,0.07366,"\int \frac{(d \sec (e+f x))^n}{(1+\sec (e+f x))^{3/2}} \, dx","Int[(d*Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2),x]","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3827, 133}"
307,1,177,0,0.2954544,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^{5/2} \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(5/2),x]","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sqrt{a \sec (e+f x)+a} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac{2 a^3 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sqrt{a \sec (e+f x)+a} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac{2 a^3 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}",1,"(2*a^3*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Sec[e + f*x]^(1 + n)*Sqrt[a + a*Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3814, 4016, 3806, 65}"
308,1,108,0,0.1446036,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(2*a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3814, 21, 3806, 65}"
309,1,48,0,0.0659588,"\int \sec ^n(e+f x) \sqrt{a+a \sec (e+f x)} \, dx","Int[Sec[e + f*x]^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",2,2,23,0.08696,1,"{3806, 65}"
310,1,61,0,0.1337539,"\int \frac{\sec ^n(e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[Sec[e + f*x]^n/Sqrt[a + a*Sec[e + f*x]],x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{a \sec (e+f x)+a}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(AppellF1[1/2, 1 - n, 1, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3828, 3825, 130, 429}"
311,1,67,0,0.1484595,"\int \frac{\sec ^n(e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[Sec[e + f*x]^n/(a + a*Sec[e + f*x])^(3/2),x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 a f \sqrt{a \sec (e+f x)+a}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 a f \sqrt{a \sec (e+f x)+a}}",1,"(AppellF1[1/2, 1 - n, 2, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*Tan[e + f*x])/(2*a*f*Sqrt[a + a*Sec[e + f*x]])","A",4,4,23,0.1739,1,"{3828, 3825, 130, 429}"
312,1,130,0,0.1612857,"\int (-\sec (e+f x))^n (a+a \sec (e+f x))^{3/2} \, dx","Int[(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 (4 n+1) \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",5,5,25,0.2000,1,"{3814, 21, 3806, 67, 65}"
313,1,70,0,0.0764576,"\int (-\sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx","Int[(-Sec[e + f*x])^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",3,3,25,0.1200,1,"{3806, 67, 65}"
314,1,75,0,0.1309893,"\int \frac{(-\sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(-Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))","A",3,3,25,0.1200,1,"{3828, 3826, 136}"
315,1,78,0,0.1445413,"\int \frac{(-\sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(-Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(a*f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))","A",3,3,25,0.1200,1,"{3828, 3826, 136}"
316,1,130,0,0.1617206,"\int (d \sec (e+f x))^n (a+a \sec (e+f x))^{3/2} \, dx","Int[(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])","A",5,5,25,0.2000,1,"{3814, 21, 3806, 67, 65}"
317,1,70,0,0.0762857,"\int (d \sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx","Int[(d*Sec[e + f*x])^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 a \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}","\frac{2 a \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])","A",3,3,25,0.1200,1,"{3806, 67, 65}"
318,1,75,0,0.1315167,"\int \frac{(d \sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Int[(d*Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
319,1,78,0,0.1450991,"\int \frac{(d \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Int[(d*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"-((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(a*f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
320,1,178,0,0.3343906,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^{5/2} \, dx","Int[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(5/2),x]","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) \sqrt{a-a \sec (e+f x)} (-\sec (e+f x))^n}{f (2 n+3)}+\frac{2 a^3 (4 n+7) \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) \sqrt{a-a \sec (e+f x)} (-\sec (e+f x))^n}{f (2 n+3)}+\frac{2 a^3 (4 n+7) \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}",1,"(2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^3*(7 + 4*n)*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 2*n))","A",4,4,26,0.1538,1,"{3814, 4016, 3806, 65}"
321,1,108,0,0.1547621,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx","Int[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"(2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]])","A",4,4,26,0.1538,1,"{3814, 21, 3806, 65}"
322,1,47,0,0.0718413,"\int (-\sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx","Int[(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]])","A",2,2,26,0.07692,1,"{3806, 65}"
323,1,58,0,0.1541175,"\int \frac{(-\sec (e+f x))^n}{\sqrt{a-a \sec (e+f x)}} \, dx","Int[(-Sec[e + f*x])^n/Sqrt[a - a*Sec[e + f*x]],x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{a-a \sec (e+f x)}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(AppellF1[1/2, 1 - n, 1, 3/2, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]])","A",4,4,26,0.1538,1,"{3828, 3825, 130, 429}"
324,1,64,0,0.1671587,"\int \frac{(-\sec (e+f x))^n}{(a-a \sec (e+f x))^{3/2}} \, dx","Int[(-Sec[e + f*x])^n/(a - a*Sec[e + f*x])^(3/2),x]","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{2 a f \sqrt{a-a \sec (e+f x)}}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{2 a f \sqrt{a-a \sec (e+f x)}}",1,"(AppellF1[1/2, 1 - n, 2, 3/2, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*Tan[e + f*x])/(2*a*f*Sqrt[a - a*Sec[e + f*x]])","A",4,4,26,0.1538,1,"{3828, 3825, 130, 429}"
325,1,130,0,0.1714049,"\int \sec ^n(e+f x) (a-a \sec (e+f x))^{3/2} \, dx","Int[Sec[e + f*x]^n*(a - a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{n+1}(e+f x) (-\sec (e+f x))^{-n} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{n+1}(e+f x) (-\sec (e+f x))^{-n} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"(2*a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]])","A",5,5,24,0.2083,1,"{3814, 21, 3806, 67, 65}"
326,1,69,0,0.0774771,"\int \sec ^n(e+f x) \sqrt{a-a \sec (e+f x)} \, dx","Int[Sec[e + f*x]^n*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2 a \sin (e+f x) (-\sec (e+f x))^{-n} \sec ^{n+1}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}","\frac{2 a \sin (e+f x) (-\sec (e+f x))^{-n} \sec ^{n+1}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]])","A",3,3,24,0.1250,1,"{3806, 67, 65}"
327,1,130,0,0.1798177,"\int (d \sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx","Int[(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2),x]","\frac{2 a^2 (4 n+1) \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}","\frac{2 a^2 (4 n+1) \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"(2*a^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]])","A",5,5,26,0.1923,1,"{3814, 21, 3806, 67, 65}"
328,1,69,0,0.0808413,"\int (d \sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx","Int[(d*Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2 a \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}","\frac{2 a \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]])","A",3,3,26,0.1154,1,"{3806, 67, 65}"
329,1,72,0,0.0640129,"\int \sec ^n(e+f x) (1+\sec (e+f x))^m \, dx","Int[Sec[e + f*x]^n*(1 + Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",2,2,19,0.1053,1,"{3825, 133}"
330,1,89,0,0.0701287,"\int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx","Int[(1 - Sec[e + f*x])^m*Sec[e + f*x]^n,x]","\frac{\sqrt{2} \tan (e+f x) (1-\sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}","\frac{\sqrt{2} \tan (e+f x) (1-\sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*(1 - Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 + Sec[e + f*x]])","A",2,2,21,0.09524,1,"{3826, 133}"
331,1,88,0,0.1093318,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f}",1,"(2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/f","A",3,3,21,0.1429,1,"{3828, 3825, 133}"
332,1,90,0,0.1200681,"\int \sec ^n(e+f x) (a-a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^n*(a - a*Sec[e + f*x])^m,x]","\frac{\sqrt{2} \tan (e+f x) (a-a \sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}","\frac{\sqrt{2} \tan (e+f x) (a-a \sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 - Sec[e + f*x], (1 - Sec[e + f*x])/2]*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 + Sec[e + f*x]])","A",3,3,22,0.1364,1,"{3828, 3826, 133}"
333,1,85,0,0.0669408,"\int (-\sec (e+f x))^n (1+\sec (e+f x))^m \, dx","Int[(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^m,x]","\frac{\sqrt{2} \tan (e+f x) (\sec (e+f x)+1)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}","\frac{\sqrt{2} \tan (e+f x) (\sec (e+f x)+1)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*(1 + Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])","A",2,2,21,0.09524,1,"{3826, 133}"
334,1,70,0,0.0654366,"\int (1-\sec (e+f x))^m (-\sec (e+f x))^n \, dx","Int[(1 - Sec[e + f*x])^m*(-Sec[e + f*x])^n,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{1-\sec (e+f x)}}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{1-\sec (e+f x)}}",1,"(2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]])","A",2,2,23,0.08696,1,"{3825, 133}"
335,1,87,0,0.1170791,"\int (-\sec (e+f x))^n (a+a \sec (e+f x))^m \, dx","Int[(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m,x]","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])","A",3,3,23,0.1304,1,"{3828, 3826, 133}"
336,1,87,0,0.1215594,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","Int[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f}",1,"(2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 + Sec[e + f*x], (1 + Sec[e + f*x])/2]*(1 - Sec[e + f*x])^(-1/2 - m)*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/f","A",3,3,24,0.1250,1,"{3828, 3825, 133}"
337,1,79,0,0.0630893,"\int (d \sec (e+f x))^n (1+\sec (e+f x))^m \, dx","Int[(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^m,x]","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,21,0.09524,1,"{3827, 133}"
338,1,79,0,0.0645752,"\int (1-\sec (e+f x))^m (d \sec (e+f x))^n \, dx","Int[(1 - Sec[e + f*x])^m*(d*Sec[e + f*x])^n,x]","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))","A",2,2,23,0.08696,1,"{3827, 133}"
339,1,95,0,0.1151435,"\int (d \sec (e+f x))^n (a+a \sec (e+f x))^m \, dx","Int[(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m,x]","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)}}","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)}}",1,"-((AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]))","A",3,3,23,0.1304,1,"{3828, 3827, 133}"
340,1,96,0,0.1189641,"\int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","Int[(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m,x]","-\frac{\tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{\sec (e+f x)+1}}",1,"-((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(1 - Sec[e + f*x])^(-1/2 - m)*(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 + Sec[e + f*x]]))","A",3,3,24,0.1250,1,"{3828, 3827, 133}"
341,1,211,0,0.3526685,"\int \sec ^4(e+f x) (a+a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^4*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} m \left(m^2+3 m+5\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2) (m+3)}+\frac{m \tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f \left(m^2+5 m+6\right)}+\frac{\tan (e+f x) \sec ^2(e+f x) (a \sec (e+f x)+a)^m}{f (m+3)}+\frac{(m+4) \tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}","\frac{2^{m+\frac{1}{2}} m \left(m^2+3 m+5\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2) (m+3)}+\frac{m \tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f \left(m^2+5 m+6\right)}+\frac{\tan (e+f x) \sec ^2(e+f x) (a \sec (e+f x)+a)^m}{f (m+3)}+\frac{(m+4) \tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}",1,"((4 + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)*(3 + m)) + (Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(3 + m)) + (2^(1/2 + m)*m*(5 + 3*m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)*(3 + m)) + (m*(a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(6 + 5*m + m^2))","A",6,6,21,0.2857,1,"{3824, 4010, 4001, 3828, 3827, 69}"
342,1,155,0,0.1949007,"\int \sec ^3(e+f x) (a+a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^3*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \left(m^2+m+1\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2)}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f \left(m^2+3 m+2\right)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f (m+2)}","\frac{2^{m+\frac{1}{2}} \left(m^2+m+1\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2)}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f \left(m^2+3 m+2\right)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f (m+2)}",1,"-(((a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(2 + 3*m + m^2))) + (2^(1/2 + m)*(1 + m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)) + ((a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(2 + m))","A",5,5,21,0.2381,1,"{3800, 4001, 3828, 3827, 69}"
343,1,107,0,0.1129581,"\int \sec ^2(e+f x) (a+a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} m \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1)}","\frac{2^{m+\frac{1}{2}} m \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1)}",1,"((a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)) + (2^(1/2 + m)*m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m))","A",4,4,21,0.1905,1,"{3798, 3828, 3827, 69}"
344,1,73,0,0.0612911,"\int \sec (e+f x) (a+a \sec (e+f x))^m \, dx","Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f}",1,"(2^(1/2 + m)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/f","A",3,3,19,0.1579,1,"{3828, 3827, 69}"
345,1,83,0,0.056996,"\int (a+a \sec (e+f x))^m \, dx","Int[(a + a*Sec[e + f*x])^m,x]","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])","A",3,3,12,0.2500,1,"{3779, 3778, 136}"
346,1,84,0,0.0963728,"\int \cos (e+f x) (a+a \sec (e+f x))^m \, dx","Int[Cos[e + f*x]*(a + a*Sec[e + f*x])^m,x]","-\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}","-\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1 + Sec[e + f*x])/2, 1 + Sec[e + f*x]]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]]))","A",3,3,19,0.1579,1,"{3828, 3827, 136}"
347,1,98,0,0.1339545,"\int (d \sec (e+f x))^{3/2} (a+a \sec (e+f x))^m \, dx","Int[(d*Sec[e + f*x])^(3/2)*(a + a*Sec[e + f*x])^m,x]","-\frac{2 \tan (e+f x) (d \sec (e+f x))^{3/2} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;\frac{5}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)}}","-\frac{2 \tan (e+f x) (d \sec (e+f x))^{3/2} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;\frac{5}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)}}",1,"(-2*AppellF1[3/2, 1/2, 1/2 - m, 5/2, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2)*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(3*f*Sqrt[1 - Sec[e + f*x]])","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
348,1,96,0,0.1178075,"\int \sqrt{d \sec (e+f x)} (a+a \sec (e+f x))^m \, dx","Int[Sqrt[d*Sec[e + f*x]]*(a + a*Sec[e + f*x])^m,x]","-\frac{2 \tan (e+f x) \sqrt{d \sec (e+f x)} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)}}","-\frac{2 \tan (e+f x) \sqrt{d \sec (e+f x)} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)}}",1,"(-2*AppellF1[1/2, 1/2, 1/2 - m, 3/2, Sec[e + f*x], -Sec[e + f*x]]*Sqrt[d*Sec[e + f*x]]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]])","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
349,1,96,0,0.1252695,"\int \frac{(a+a \sec (e+f x))^m}{\sqrt{d \sec (e+f x)}} \, dx","Int[(a + a*Sec[e + f*x])^m/Sqrt[d*Sec[e + f*x]],x]","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)} \sqrt{d \sec (e+f x)}}","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(2*AppellF1[-1/2, 1/2, 1/2 - m, 1/2, Sec[e + f*x], -Sec[e + f*x]]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]]*Sqrt[d*Sec[e + f*x]])","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
350,1,98,0,0.1355674,"\int \frac{(a+a \sec (e+f x))^m}{(d \sec (e+f x))^{3/2}} \, dx","Int[(a + a*Sec[e + f*x])^m/(d*Sec[e + f*x])^(3/2),x]","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;-\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)} (d \sec (e+f x))^{3/2}}","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;-\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)} (d \sec (e+f x))^{3/2}}",1,"(2*AppellF1[-3/2, 1/2, 1/2 - m, -1/2, Sec[e + f*x], -Sec[e + f*x]]*(1 + Sec[e + f*x])^(-1/2 - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(3*f*Sqrt[1 - Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2))","A",3,3,25,0.1200,1,"{3828, 3827, 133}"
351,1,111,0,0.0902441,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[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",7,5,21,0.2381,1,"{4225, 2748, 2635, 2639, 2641}"
352,1,87,0,0.080015,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[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",6,5,21,0.2381,1,"{4225, 2748, 2635, 2641, 2639}"
353,1,61,0,0.0693064,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[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",5,5,21,0.2381,1,"{4225, 2748, 2639, 2635, 2641}"
354,1,35,0,0.0587835,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[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",4,4,21,0.1905,1,"{4225, 2748, 2641, 2639}"
355,1,57,0,0.0673231,"\int \frac{a+a \sec (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Sec[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 \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",5,5,21,0.2381,1,"{4225, 2748, 2636, 2639, 2641}"
356,1,83,0,0.0790063,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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)}{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",6,5,21,0.2381,1,"{4225, 2748, 2636, 2641, 2639}"
357,1,111,0,0.0907566,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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{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",7,5,21,0.2381,1,"{4225, 2748, 2636, 2639, 2641}"
358,1,135,0,0.1015162,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])/Cos[c + d*x]^(7/2),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{10 a \sin (c+d x)}{21 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{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d 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{10 a \sin (c+d x)}{21 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{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{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) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (10*a*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,5,21,0.2381,1,"{4225, 2748, 2636, 2641, 2639}"
359,1,147,0,0.1742665,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[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",10,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2641, 4045, 2639}"
360,1,121,0,0.1576815,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[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",9,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2639, 4045, 2641}"
361,1,95,0,0.1432989,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[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",8,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2641, 4045, 2639}"
362,1,67,0,0.1280576,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^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) \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",7,6,23,0.2609,1,"{4264, 3788, 3771, 2639, 4045, 2641}"
363,1,44,0,0.1094876,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^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",5,5,23,0.2174,1,"{4264, 3788, 3771, 2641, 4043}"
364,1,91,0,0.1385962,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Sec[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)}{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",8,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2639, 4046, 2641}"
365,1,121,0,0.1535813,"\int \frac{(a+a \sec (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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)}{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",9,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2641, 4046, 2639}"
366,1,147,0,0.1729435,"\int \frac{(a+a \sec (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[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)}{7 d}-\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^2 \sin (c+d x)}{7 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d 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{8 a^2 \sin (c+d x)}{7 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-12*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*EllipticF[(c + d*x)/2, 2])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (4*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(3/2)) + (12*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",10,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2639, 4046, 2641}"
367,1,147,0,0.251881,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[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",17,6,23,0.2609,1,"{4264, 3791, 3769, 3771, 2639, 2641}"
368,1,121,0,0.2196711,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[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",15,6,23,0.2609,1,"{4264, 3791, 3769, 3771, 2641, 2639}"
369,1,91,0,0.1857393,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3,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",13,6,23,0.2609,1,"{4264, 3791, 3769, 3771, 2639, 2641}"
370,1,91,0,0.1967503,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3,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",13,7,23,0.3043,1,"{4264, 3791, 3769, 3771, 2641, 2639, 3768}"
371,1,91,0,0.1941527,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3,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",13,6,23,0.2609,1,"{4264, 3791, 3771, 2639, 2641, 3768}"
372,1,117,0,0.2232261,"\int \frac{(a+a \sec (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Sec[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)}{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",15,6,23,0.2609,1,"{4264, 3791, 3771, 2641, 3768, 2639}"
373,1,147,0,0.2386766,"\int \frac{(a+a \sec (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^3/Cos[c + d*x]^(3/2),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",17,6,23,0.2609,1,"{4264, 3791, 3768, 3771, 2639, 2641}"
374,1,128,0,0.1757281,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Sec[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{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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}","-\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{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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}",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]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{4264, 3819, 3787, 3769, 3771, 2639, 2641}"
375,1,100,0,0.1613661,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Sec[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{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \sec (c+d x)+a)}","\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{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \sec (c+d x)+a)}",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) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,7,23,0.3043,1,"{4264, 3819, 3787, 3769, 3771, 2641, 2639}"
376,1,72,0,0.1470755,"\int \frac{\sqrt{\cos (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Sec[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)}{d \sqrt{\cos (c+d x)} (a \sec (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)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"(3*EllipticE[(c + d*x)/2, 2])/(a*d) - EllipticF[(c + d*x)/2, 2]/(a*d) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{4264, 3819, 3787, 3771, 2639, 2641}"
377,1,70,0,0.1413765,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[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)}{d \sqrt{\cos (c+d x)} (a \sec (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)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"-(EllipticE[(c + d*x)/2, 2]/(a*d)) + EllipticF[(c + d*x)/2, 2]/(a*d) + Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{4264, 3820, 3787, 3771, 2639, 2641}"
378,1,70,0,0.1437337,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[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)}{d \sqrt{\cos (c+d x)} (a \sec (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)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"EllipticE[(c + d*x)/2, 2]/(a*d) + EllipticF[(c + d*x)/2, 2]/(a*d) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,23,0.2609,1,"{4264, 3818, 3787, 3771, 2639, 2641}"
379,1,96,0,0.1565398,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[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 \cos ^{\frac{3}{2}}(c+d x) (a \sec (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 \cos ^{\frac{3}{2}}(c+d x) (a \sec (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*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",8,7,23,0.3043,1,"{4264, 3818, 3787, 3771, 2641, 3768, 2639}"
380,1,124,0,0.1748899,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[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{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{\sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}","\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{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{\sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}",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]^(5/2)*(a + a*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{4264, 3818, 3787, 3768, 3771, 2639, 2641}"
381,1,160,0,0.2838125,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Sec[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{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{3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (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{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{3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (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]^(3/2)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",10,8,23,0.3478,1,"{4264, 3817, 4020, 3787, 3769, 3771, 2639, 2641}"
382,1,138,0,0.2672157,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Sec[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{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \sec (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{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \sec (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*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,8,23,0.3478,1,"{4264, 3817, 4020, 3787, 3769, 3771, 2641, 2639}"
383,1,112,0,0.2422247,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Sec[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)}{3 a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \sec (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)}{3 a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \sec (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*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{4264, 3817, 4020, 3787, 3771, 2639, 2641}"
384,1,109,0,0.2405849,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[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)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (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)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (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) + Sin[c + d*x]/(a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{4264, 3817, 4019, 3787, 3771, 2639, 2641}"
385,1,57,0,0.0992939,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[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)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (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)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"EllipticF[(c + d*x)/2, 2]/(3*a^2*d) + Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",5,5,23,0.2174,1,"{4264, 3815, 21, 3771, 2641}"
386,1,109,0,0.2424947,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[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)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (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)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (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) - Sin[c + d*x]/(a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",8,7,23,0.3043,1,"{4264, 3816, 4019, 3787, 3771, 2639, 2641}"
387,1,136,0,0.2627844,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[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 \cos ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (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 \cos ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (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*Cos[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2)","A",9,8,23,0.3478,1,"{4264, 3816, 4019, 3787, 3771, 2641, 3768, 2639}"
388,1,162,0,0.2882127,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[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{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{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{5}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (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{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{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{5}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (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]^(5/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2)","A",10,8,23,0.3478,1,"{4264, 3816, 4019, 3787, 3768, 3771, 2639, 2641}"
389,1,207,0,0.4284364,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Sec[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{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{63 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}","-\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{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{63 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}",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]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (63*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",11,8,23,0.3478,1,"{4264, 3817, 4020, 3787, 3769, 3771, 2639, 2641}"
390,1,181,0,0.3963543,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Sec[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{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (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{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (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*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))","A",10,8,23,0.3478,1,"{4264, 3817, 4020, 3787, 3769, 3771, 2641, 2639}"
391,1,155,0,0.3695697,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^3} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Sec[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)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \sec (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{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \sec (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) - Sin[c + d*x]/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{4264, 3817, 4020, 3787, 3771, 2639, 2641}"
392,1,155,0,0.3847333,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[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{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)} (a \sec (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{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)} (a \sec (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) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) + (2*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{4264, 3817, 4019, 4020, 3787, 3771, 2639, 2641}"
393,1,155,0,0.3780317,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[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)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}","\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)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"-EllipticE[(c + d*x)/2, 2]/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(6*a^3*d) + Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - Sin[c + d*x]/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{4264, 3815, 4019, 4020, 3787, 3771, 2639, 2641}"
394,1,155,0,0.3819504,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[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)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}","\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)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"EllipticE[(c + d*x)/2, 2]/(10*a^3*d) + EllipticF[(c + d*x)/2, 2]/(6*a^3*d) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (4*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{4264, 3816, 4019, 4020, 3787, 3771, 2639, 2641}"
395,1,155,0,0.3904249,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[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)}{10 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{5 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (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)}{10 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{5 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (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) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (9*Sin[c + d*x])/(10*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,7,23,0.3043,1,"{4264, 3816, 4019, 3787, 3771, 2639, 2641}"
396,1,181,0,0.4062744,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[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 \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (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 \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (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*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))","A",10,8,23,0.3478,1,"{4264, 3816, 4019, 3787, 3771, 2641, 3768, 2639}"
397,1,207,0,0.4341421,"\int \frac{1}{\cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(11/2)*(a + a*Sec[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{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{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a \sec (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{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{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a \sec (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]^(9/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Cos[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x]))","A",11,8,23,0.3478,1,"{4264, 3816, 4019, 3787, 3768, 3771, 2639, 2641}"
398,1,153,0,0.2977716,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\cos (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\cos (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(32*a*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (12*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",5,3,25,0.1200,1,"{4264, 3805, 3804}"
399,1,115,0,0.2322765,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",4,3,25,0.1200,1,"{4264, 3805, 3804}"
400,1,77,0,0.1735425,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{4264, 3805, 3804}"
401,1,36,0,0.1099592,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{4264, 3804}"
402,1,57,0,0.1161871,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{2 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{2 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d","A",3,3,25,0.1200,1,"{4264, 3801, 215}"
403,1,92,0,0.1746064,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{a \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{4264, 3803, 3801, 215}"
404,1,136,0,0.229755,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{3 a \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{3 a \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(3*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",5,4,25,0.1600,1,"{4264, 3803, 3801, 215}"
405,1,161,0,0.3114556,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(208*a^2*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (104*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (26*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,25,0.2000,1,"{4264, 3813, 21, 3805, 3804}"
406,1,116,0,0.2424473,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{5 d}","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{5 d}",1,"(8*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",4,4,25,0.1600,1,"{4264, 3812, 3809, 3804}"
407,1,79,0,0.1761762,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(8*a^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",3,3,25,0.1200,1,"{4264, 3809, 3804}"
408,1,96,0,0.1829579,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4264, 3813, 21, 3801, 215}"
409,1,95,0,0.1808456,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a^2 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(3*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4264, 3814, 21, 3801, 215}"
410,1,140,0,0.2425848,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{7 a^2 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{7 a^2 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{7 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(7*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (7*a^2*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4264, 3814, 21, 3803, 3801, 215}"
411,1,180,0,0.3024727,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\frac{11 a^2 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}","\frac{11 a^2 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{11 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}",1,"(11*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4264, 3814, 21, 3803, 3801, 215}"
412,1,201,0,0.4061007,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}+\frac{38 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}+\frac{38 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(1168*a^3*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (584*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (146*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (38*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",6,5,25,0.2000,1,"{4264, 3813, 4015, 3805, 3804}"
413,1,156,0,0.299475,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(64*a^3*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",5,4,25,0.1600,1,"{4264, 3812, 3809, 3804}"
414,1,119,0,0.2313806,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(64*a^3*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",4,3,25,0.1200,1,"{4264, 3809, 3804}"
415,1,138,0,0.2821255,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,25,0.2000,1,"{4264, 3813, 4015, 3801, 215}"
416,1,132,0,0.2807002,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}","\frac{a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{5 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(5*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,25,0.2000,1,"{4264, 3814, 4015, 3801, 215}"
417,1,140,0,0.2798871,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{9 a^3 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{19 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}","\frac{9 a^3 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{19 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(19*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))","A",5,5,25,0.2000,1,"{4264, 3814, 4016, 3801, 215}"
418,1,180,0,0.3425482,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{25 a^3 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{13 a^3 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{25 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}","\frac{25 a^3 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{13 a^3 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{25 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}",1,"(25*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (13*a^3*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (25*a^3*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))","A",6,6,25,0.2400,1,"{4264, 3814, 4016, 3803, 3801, 215}"
419,1,220,0,0.4030763,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{163 a^3 \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{163 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}","\frac{163 a^3 \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{163 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}",1,"(163*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (17*a^3*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))","A",7,6,25,0.2400,1,"{4264, 3814, 4016, 3803, 3801, 215}"
420,1,189,0,0.4137192,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4264, 3823, 4022, 4013, 3808, 206}"
421,1,151,0,0.2826252,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4264, 3823, 4013, 3808, 206}"
422,1,113,0,0.1701657,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{4264, 3812, 3808, 206}"
423,1,76,0,0.11586,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)","A",3,3,25,0.1200,1,"{4264, 3808, 206}"
424,1,135,0,0.2383141,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)","A",6,6,25,0.2400,1,"{4264, 3821, 3801, 215, 3808, 206}"
425,1,168,0,0.3400521,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{4264, 3822, 4023, 3808, 206, 3801, 215}"
426,1,211,0,0.4750038,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}","-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}",1,"(7*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{4264, 3822, 4021, 4023, 3808, 206, 3801, 215}"
427,1,237,0,0.5940952,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{15 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{49 \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}","-\frac{15 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{49 \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-15*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + (9*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,25,0.2400,1,"{4264, 3817, 4022, 4013, 3808, 206}"
428,1,197,0,0.4462498,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}","\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(11*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (19*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4264, 3817, 4022, 4013, 3808, 206}"
429,1,157,0,0.3066247,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}","-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(-7*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (5*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{4264, 3817, 4013, 3808, 206}"
430,1,117,0,0.1872287,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{4264, 3811, 3808, 206}"
431,1,117,0,0.1877415,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{4264, 3810, 3808, 206}"
432,1,174,0,0.3558038,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}","-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",7,7,25,0.2800,1,"{4264, 3816, 4023, 3808, 206, 3801, 215}"
433,1,214,0,0.4918323,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}","\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(-3*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + (9*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{4264, 3816, 4021, 4023, 3808, 206, 3801, 215}"
434,1,237,0,0.5955815,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{95 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}","\frac{95 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(163*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (17*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (299*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (95*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{4264, 3817, 4020, 4022, 4013, 3808, 206}"
435,1,197,0,0.4583663,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{49 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{75 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}","\frac{49 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{75 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(-75*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4264, 3817, 4020, 4013, 3808, 206}"
436,1,157,0,0.3150355,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{19 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}","\frac{19 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(19*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - (9*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{4264, 3817, 4012, 3808, 206}"
437,1,157,0,0.2597461,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}","\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + (5*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{4264, 3811, 3810, 3808, 206}"
438,1,157,0,0.2594962,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,4,25,0.1600,1,"{4264, 3810, 3808, 206}"
439,1,214,0,0.502705,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{43 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{11 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}","-\frac{43 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{11 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) - (11*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",8,8,25,0.3200,1,"{4264, 3816, 4019, 4023, 3808, 206, 3801, 215}"
440,1,254,0,0.645607,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{35 \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{115 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{15 \sin (c+d x)}{16 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}","\frac{35 \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{115 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{15 \sin (c+d x)}{16 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(-5*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + (115*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) - (15*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{4264, 3816, 4019, 4021, 4023, 3808, 206, 3801, 215}"
441,1,244,0,0.3997343,"\int (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx","Int[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3,x]","-\frac{a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) (d \cos (e+f x))^n}{f (2-n)}","-\frac{a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) (d \cos (e+f x))^n}{f (2-n)}",1,"-((a^3*(7 - 4*n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a^3*(1 - 4*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + ((d*Cos[e + f*x])^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n))","A",8,6,23,0.2609,1,"{4264, 3814, 3997, 3787, 3772, 2643}"
442,1,179,0,0.2263405,"\int (d \cos (e+f x))^n (a+a \sec (e+f x))^2 \, dx","Int[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^2,x]","-\frac{2 a^2 \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a^2 (1-2 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}","-\frac{2 a^2 \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a^2 (1-2 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}",1,"(-2*a^2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2]) - (a^2*(1 - 2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n))","A",7,5,23,0.2174,1,"{4264, 3788, 3772, 2643, 4046}"
443,1,132,0,0.1178298,"\int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx","Int[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x]),x]","-\frac{a \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}","-\frac{a \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}",1,"-((a*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*(1 + n)*Sqrt[Sin[e + f*x]^2])","A",5,4,21,0.1905,1,"{4225, 16, 2748, 2643}"
444,1,178,0,0.2444987,"\int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, dx","Int[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]),x]","-\frac{\sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(n+1) \sin (e+f x) \cos ^2(e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(e+f x)\right)}{a f (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) (d \cos (e+f x))^n}{f (a \sec (e+f x)+a)}","-\frac{\sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(n+1) \sin (e+f x) \cos ^2(e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(e+f x)\right)}{a f (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) (d \cos (e+f x))^n}{f (a \sec (e+f x)+a)}",1,"((d*Cos[e + f*x])^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 + n)*Cos[e + f*x]^2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(2 + n)*Sqrt[Sin[e + f*x]^2])","A",7,5,23,0.2174,1,"{4264, 3820, 3787, 3772, 2643}"
445,1,215,0,0.4117792,"\int \frac{(d \cos (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","Int[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x])^2,x]","\frac{2 (n+2) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(2 n+3) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (n+2) \tan (e+f x) (d \cos (e+f x))^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) (d \cos (e+f x))^n}{3 f (a \sec (e+f x)+a)^2}","\frac{2 (n+2) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(2 n+3) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (n+2) \tan (e+f x) (d \cos (e+f x))^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) (d \cos (e+f x))^n}{3 f (a \sec (e+f x)+a)^2}",1,"(2*(2 + n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - ((3 + 2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - (2*(2 + n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - ((d*Cos[e + f*x])^n*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)","A",8,6,23,0.2609,1,"{4264, 3817, 4020, 3787, 3772, 2643}"
446,1,85,0,0.0665066,"\int \sec ^4(c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x]),x]","\frac{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 ^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) + (a*Tan[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{3787, 3767, 3768, 3770}"
447,1,63,0,0.052903,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x]),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 ^3(c+d x)}{3 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 ^3(c+d x)}{3 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) + (b*Tan[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{3787, 3768, 3770, 3767}"
448,1,47,0,0.0492891,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x]),x]","\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 (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",5,5,19,0.2632,1,"{3787, 3767, 8, 3768, 3770}"
449,1,24,0,0.0260089,"\int \sec (c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d","A",4,4,17,0.2353,1,"{3787, 3770, 3767, 8}"
450,1,16,0,0.0081961,"\int (a+b \sec (c+d x)) \, dx","Int[a + b*Sec[c + d*x],x]","a x+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","a x+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*x + (b*ArcTanh[Sin[c + d*x]])/d","A",2,1,10,0.1000,1,"{3770}"
451,1,15,0,0.0231894,"\int \cos (c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+b x","\frac{a \sin (c+d x)}{d}+b x",1,"b*x + (a*Sin[c + d*x])/d","A",3,3,17,0.1765,1,"{3787, 2637, 8}"
452,1,38,0,0.0377009,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x]),x]","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}+\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 (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)","A",4,4,19,0.2105,1,"{3787, 2635, 8, 2637}"
453,1,54,0,0.0458303,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[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 (c+d x)}{2 d}+\frac{b x}{2}","-\frac{a \sin ^3(c+d x)}{3 d}+\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*Sin[c + d*x]^3)/(3*d)","A",5,4,19,0.2105,1,"{3787, 2633, 2635, 8}"
454,1,76,0,0.056168,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[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 ^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 ^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) - (b*Sin[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{3787, 2635, 8, 2633}"
455,1,92,0,0.0582848,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[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 ^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 ^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 ^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) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",6,4,19,0.2105,1,"{3787, 2633, 2635, 8}"
456,1,135,0,0.1063546,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","\frac{\left(5 a^2+4 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a^2+4 b^2\right) \tan (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{b^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}","\frac{\left(5 a^2+4 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a^2+4 b^2\right) \tan (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{b^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) + ((5*a^2 + 4*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) + (b^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*a^2 + 4*b^2)*Tan[c + d*x]^3)/(15*d)","A",7,5,21,0.2381,1,"{3788, 3768, 3770, 4046, 3767}"
457,1,110,0,0.0941703,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","\frac{\left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}","\frac{\left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Tan[c + d*x])/d + ((4*a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^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,"{3788, 3767, 4046, 3768, 3770}"
458,1,80,0,0.0901502,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","\frac{\left(3 a^2+2 b^2\right) \tan (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{b^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}","\frac{\left(3 a^2+2 b^2\right) \tan (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{b^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*b*ArcTanh[Sin[c + d*x]])/d + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (b^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,21,0.2857,1,"{3788, 3768, 3770, 4046, 3767, 8}"
459,1,59,0,0.0536586,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"((2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Tan[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,5,19,0.2632,1,"{3788, 3767, 8, 4046, 3770}"
460,1,33,0,0.0256904,"\int (a+b \sec (c+d x))^2 \, dx","Int[(a + b*Sec[c + d*x])^2,x]","a^2 x+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","a^2 x+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"a^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (b^2*Tan[c + d*x])/d","A",4,4,12,0.3333,1,"{3773, 3770, 3767, 8}"
461,1,33,0,0.0554316,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \sin (c+d x)}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \sin (c+d x)}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"2*a*b*x + (b^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d","A",4,4,19,0.2105,1,"{3788, 8, 4045, 3770}"
462,1,50,0,0.0657883,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","\frac{1}{2} x \left(a^2+2 b^2\right)+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a b \sin (c+d x)}{d}","\frac{1}{2} x \left(a^2+2 b^2\right)+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a b \sin (c+d x)}{d}",1,"((a^2 + 2*b^2)*x)/2 + (2*a*b*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",4,4,21,0.1905,1,"{3788, 2637, 4045, 8}"
463,1,58,0,0.089153,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{d}+a b x","\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{d}+a b x",1,"a*b*x + ((a^2 + b^2)*Sin[c + d*x])/d + (a*b*Cos[c + d*x]*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d)","A",6,5,21,0.2381,1,"{3788, 2635, 8, 4044, 3013}"
464,1,101,0,0.0878394,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+4 b^2\right)+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}","\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+4 b^2\right)+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}",1,"((3*a^2 + 4*b^2)*x)/8 + (2*a*b*Sin[c + d*x])/d + ((3*a^2 + 4*b^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*b*Sin[c + d*x]^3)/(3*d)","A",6,5,21,0.2381,1,"{3788, 2633, 4045, 2635, 8}"
465,1,111,0,0.1224427,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2,x]","-\frac{\left(2 a^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^2 \sin ^5(c+d x)}{5 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{\left(2 a^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^2 \sin ^5(c+d x)}{5 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}",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) - ((2*a^2 + b^2)*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^5)/(5*d)","A",8,6,21,0.2857,1,"{3788, 2635, 8, 4044, 3013, 373}"
466,1,189,0,0.3122819,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","-\frac{\left(-52 a^2 b^2+3 a^4-16 b^4\right) \tan (c+d x)}{30 b d}+\frac{a \left(4 a^2+9 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left(3 a^2-16 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}","-\frac{\left(-52 a^2 b^2+3 a^4-16 b^4\right) \tan (c+d x)}{30 b d}+\frac{a \left(4 a^2+9 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left(3 a^2-16 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}",1,"(a*(4*a^2 + 9*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Tan[c + d*x])/(30*b*d) - (a*(6*a^2 - 71*b^2)*Sec[c + d*x]*Tan[c + d*x])/(120*d) - ((3*a^2 - 16*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) - (a*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + ((a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)","A",8,7,21,0.3333,1,"{3840, 4002, 3997, 3787, 3770, 3767, 8}"
467,1,130,0,0.198026,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","\frac{a \left(a^2+4 b^2\right) \tan (c+d x)}{2 d}+\frac{3 b \left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(2 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}","\frac{a \left(a^2+4 b^2\right) \tan (c+d x)}{2 d}+\frac{3 b \left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(2 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(3*b*(4*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(a^2 + 4*b^2)*Tan[c + d*x])/(2*d) + (b*(2*a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + ((a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,7,21,0.3333,1,"{3835, 4002, 3997, 3787, 3770, 3767, 8}"
468,1,99,0,0.1310792,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^3,x]","\frac{2 b \left(4 a^2+b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a b^2 \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}","\frac{2 b \left(4 a^2+b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a b^2 \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(a*(2*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*b*(4*a^2 + b^2)*Tan[c + d*x])/(3*d) + (5*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,6,19,0.3158,1,"{3830, 3997, 3787, 3770, 3767, 8}"
469,1,73,0,0.0485526,"\int (a+b \sec (c+d x))^3 \, dx","Int[(a + b*Sec[c + d*x])^3,x]","\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x+\frac{5 a b^2 \tan (c+d x)}{2 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d}","\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x+\frac{5 a b^2 \tan (c+d x)}{2 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"a^3*x + (b*(6*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a*b^2*Tan[c + d*x])/(2*d) + (b^2*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",5,4,12,0.3333,1,"{3782, 3770, 3767, 8}"
470,1,67,0,0.1118612,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^3,x]","\frac{a \left(a^2-b^2\right) \sin (c+d x)}{d}+3 a^2 b x+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))}{d}","\frac{a \left(a^2-b^2\right) \sin (c+d x)}{d}+3 a^2 b x+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))}{d}",1,"3*a^2*b*x + (3*a*b^2*ArcTanh[Sin[c + d*x]])/d + (a*(a^2 - b^2)*Sin[c + d*x])/d + (b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/d","A",5,5,19,0.2632,1,"{3842, 4047, 8, 4045, 3770}"
471,1,79,0,0.1193983,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","\frac{1}{2} a x \left(a^2+6 b^2\right)+\frac{5 a^2 b \sin (c+d x)}{2 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))}{2 d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{1}{2} a x \left(a^2+6 b^2\right)+\frac{5 a^2 b \sin (c+d x)}{2 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))}{2 d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*(a^2 + 6*b^2)*x)/2 + (b^3*ArcTanh[Sin[c + d*x]])/d + (5*a^2*b*Sin[c + d*x])/(2*d) + (a^2*Cos[c + d*x]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",5,5,21,0.2381,1,"{3841, 4047, 8, 4045, 3770}"
472,1,100,0,0.1499884,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","\frac{a \left(2 a^2+9 b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} b x \left(3 a^2+2 b^2\right)+\frac{7 a^2 b \sin (c+d x) \cos (c+d x)}{6 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))}{3 d}","\frac{a \left(2 a^2+9 b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} b x \left(3 a^2+2 b^2\right)+\frac{7 a^2 b \sin (c+d x) \cos (c+d x)}{6 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))}{3 d}",1,"(b*(3*a^2 + 2*b^2)*x)/2 + (a*(2*a^2 + 9*b^2)*Sin[c + d*x])/(3*d) + (7*a^2*b*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",5,5,21,0.2381,1,"{3841, 4047, 2637, 4045, 8}"
473,1,123,0,0.1832816,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3,x]","\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{4 d}+\frac{3 a \left(a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2+4 b^2\right)-\frac{3 a^2 b \sin ^3(c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))}{4 d}","\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{4 d}+\frac{3 a \left(a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2+4 b^2\right)-\frac{3 a^2 b \sin ^3(c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))}{4 d}",1,"(3*a*(a^2 + 4*b^2)*x)/8 + (b*(11*a^2 + 4*b^2)*Sin[c + d*x])/(4*d) + (3*a*(a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])*Sin[c + d*x])/(4*d) - (3*a^2*b*Sin[c + d*x]^3)/(4*d)","A",7,6,21,0.2857,1,"{3841, 4047, 2635, 8, 4044, 3013}"
474,1,160,0,0.1919362,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3,x]","-\frac{a \left(4 a^2+15 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \sin (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(9 a^2+4 b^2\right)+\frac{11 a^2 b \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))}{5 d}","-\frac{a \left(4 a^2+15 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \sin (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(9 a^2+4 b^2\right)+\frac{11 a^2 b \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(b*(9*a^2 + 4*b^2)*x)/8 + (a*(4*a^2 + 15*b^2)*Sin[c + d*x])/(5*d) + (b*(9*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (11*a^2*b*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a^2*Cos[c + d*x]^4*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a*(4*a^2 + 15*b^2)*Sin[c + d*x]^3)/(15*d)","A",7,6,21,0.2857,1,"{3841, 4047, 2633, 4045, 2635, 8}"
475,1,185,0,0.2321064,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]","-\frac{b \left(5 a^2+b^2\right) \sin ^3(c+d x)}{3 d}+\frac{b \left(17 a^2+6 b^2\right) \sin (c+d x)}{6 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(5 a^2+18 b^2\right)+\frac{13 a^2 b \sin ^5(c+d x)}{30 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))}{6 d}","-\frac{b \left(5 a^2+b^2\right) \sin ^3(c+d x)}{3 d}+\frac{b \left(17 a^2+6 b^2\right) \sin (c+d x)}{6 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(5 a^2+18 b^2\right)+\frac{13 a^2 b \sin ^5(c+d x)}{30 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))}{6 d}",1,"(a*(5*a^2 + 18*b^2)*x)/16 + (b*(17*a^2 + 6*b^2)*Sin[c + d*x])/(6*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])*Sin[c + d*x])/(6*d) - (b*(5*a^2 + b^2)*Sin[c + d*x]^3)/(3*d) + (13*a^2*b*Sin[c + d*x]^5)/(30*d)","A",9,7,21,0.3333,1,"{3841, 4047, 2635, 8, 4044, 3013, 373}"
476,1,244,0,0.4502679,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]","-\frac{a \left(-121 a^2 b^2+4 a^4-128 b^4\right) \tan (c+d x)}{60 b d}+\frac{\left(36 a^2 b^2+8 a^4+5 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left(4 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}-\frac{\left(-178 a^2 b^2+8 a^4-75 b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}","-\frac{a \left(-121 a^2 b^2+4 a^4-128 b^4\right) \tan (c+d x)}{60 b d}+\frac{\left(36 a^2 b^2+8 a^4+5 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left(4 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}-\frac{\left(-178 a^2 b^2+8 a^4-75 b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}",1,"((8*a^4 + 36*a^2*b^2 + 5*b^4)*ArcTanh[Sin[c + d*x]])/(16*d) - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Tan[c + d*x])/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*Sec[c + d*x]*Tan[c + d*x])/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) - (a*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + ((a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)","A",9,7,21,0.3333,1,"{3840, 4002, 3997, 3787, 3770, 3767, 8}"
477,1,179,0,0.3018705,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4,x]","\frac{2 \left(28 a^2 b^2+3 a^4+4 b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}","\frac{2 \left(28 a^2 b^2+3 a^4+4 b^4\right) \tan (c+d x)}{15 d}+\frac{a b \left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}",1,"(a*b*(4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(15*d) + (a*b*(6*a^2 + 29*b^2)*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*a^2 + 4*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + ((a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)","A",8,7,21,0.3333,1,"{3835, 4002, 3997, 3787, 3770, 3767, 8}"
478,1,146,0,0.242574,"\int \sec (c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^4,x]","\frac{a b \left(19 a^2+16 b^2\right) \tan (c+d x)}{6 d}+\frac{\left(24 a^2 b^2+8 a^4+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{7 a b \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}","\frac{a b \left(19 a^2+16 b^2\right) \tan (c+d x)}{6 d}+\frac{\left(24 a^2 b^2+8 a^4+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{7 a b \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}",1,"((8*a^4 + 24*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b*(19*a^2 + 16*b^2)*Tan[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (7*a*b*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,7,19,0.3684,1,"{3830, 4002, 3997, 3787, 3770, 3767, 8}"
479,1,107,0,0.1157478,"\int (a+b \sec (c+d x))^4 \, dx","Int[(a + b*Sec[c + d*x])^4,x]","\frac{b^2 \left(17 a^2+2 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+a^4 x+\frac{4 a b^3 \tan (c+d x) \sec (c+d x)}{3 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}","\frac{b^2 \left(17 a^2+2 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+a^4 x+\frac{4 a b^3 \tan (c+d x) \sec (c+d x)}{3 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"a^4*x + (2*a*b*(2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/d + (b^2*(17*a^2 + 2*b^2)*Tan[c + d*x])/(3*d) + (4*a*b^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,5,12,0.4167,1,"{3782, 4048, 3770, 3767, 8}"
480,1,104,0,0.21183,"\int \cos (c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^4,x]","\frac{a^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{b^2 \left(12 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+4 a^3 b x+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}","\frac{a^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{b^2 \left(12 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+4 a^3 b x+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}",1,"4*a^3*b*x + (b^2*(12*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*a^2 - b^2)*Sin[c + d*x])/(2*d) + (b^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (3*a*b^3*Tan[c + d*x])/d","A",6,6,19,0.3158,1,"{3842, 4076, 4047, 8, 4045, 3770}"
481,1,108,0,0.2174044,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4,x]","-\frac{b^2 \left(a^2-2 b^2\right) \tan (c+d x)}{2 d}+\frac{1}{2} a^2 x \left(a^2+12 b^2\right)+\frac{3 a^3 b \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{4 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{b^2 \left(a^2-2 b^2\right) \tan (c+d x)}{2 d}+\frac{1}{2} a^2 x \left(a^2+12 b^2\right)+\frac{3 a^3 b \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{4 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a^2*(a^2 + 12*b^2)*x)/2 + (4*a*b^3*ArcTanh[Sin[c + d*x]])/d + (3*a^3*b*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a^2 - 2*b^2)*Tan[c + d*x])/(2*d)","A",6,6,21,0.2857,1,"{3841, 4076, 4047, 8, 4045, 3770}"
482,1,115,0,0.2397845,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]","\frac{a^2 \left(2 a^2+17 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(a^2+2 b^2\right)+\frac{4 a^3 b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \left(2 a^2+17 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(a^2+2 b^2\right)+\frac{4 a^3 b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{d}",1,"2*a*b*(a^2 + 2*b^2)*x + (b^4*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2 + 17*b^2)*Sin[c + d*x])/(3*d) + (4*a^3*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,21,0.2857,1,"{3841, 4074, 4047, 8, 4045, 3770}"
483,1,145,0,0.3128649,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4,x]","\frac{4 a b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(24 a^2 b^2+3 a^4+8 b^4\right)+\frac{5 a^3 b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}","\frac{4 a b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(24 a^2 b^2+3 a^4+8 b^4\right)+\frac{5 a^3 b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"((3*a^4 + 24*a^2*b^2 + 8*b^4)*x)/8 + (4*a*b*(2*a^2 + 3*b^2)*Sin[c + d*x])/(3*d) + (a^2*(3*a^2 + 22*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a^3*b*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,21,0.2857,1,"{3841, 4074, 4047, 2637, 4045, 8}"
484,1,173,0,0.3517319,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4,x]","-\frac{a^2 \left(4 a^2+27 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{\left(29 a^2 b^2+4 a^4+5 b^4\right) \sin (c+d x)}{5 d}+\frac{a b \left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a b x \left(3 a^2+4 b^2\right)+\frac{3 a^3 b \sin (c+d x) \cos ^3(c+d x)}{5 d}+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}","-\frac{a^2 \left(4 a^2+27 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{\left(29 a^2 b^2+4 a^4+5 b^4\right) \sin (c+d x)}{5 d}+\frac{a b \left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a b x \left(3 a^2+4 b^2\right)+\frac{3 a^3 b \sin (c+d x) \cos ^3(c+d x)}{5 d}+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"(a*b*(3*a^2 + 4*b^2)*x)/2 + ((4*a^4 + 29*a^2*b^2 + 5*b^4)*Sin[c + d*x])/(5*d) + (a*b*(3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a^3*b*Cos[c + d*x]^3*Sin[c + d*x])/(5*d) + (a^2*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a^2*(4*a^2 + 27*b^2)*Sin[c + d*x]^3)/(15*d)","A",8,7,21,0.3333,1,"{3841, 4074, 4047, 2635, 8, 4044, 3013}"
485,1,213,0,0.3800106,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4,x]","-\frac{4 a b \left(4 a^2+5 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \sin (c+d x)}{5 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(36 a^2 b^2+5 a^4+8 b^4\right)+\frac{7 a^3 b \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^2}{6 d}","-\frac{4 a b \left(4 a^2+5 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \sin (c+d x)}{5 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(36 a^2 b^2+5 a^4+8 b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(36 a^2 b^2+5 a^4+8 b^4\right)+\frac{7 a^3 b \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^2}{6 d}",1,"((5*a^4 + 36*a^2*b^2 + 8*b^4)*x)/16 + (4*a*b*(4*a^2 + 5*b^2)*Sin[c + d*x])/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(5*a^2 + 32*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (7*a^3*b*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (a^2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) - (4*a*b*(4*a^2 + 5*b^2)*Sin[c + d*x]^3)/(15*d)","A",8,7,21,0.3333,1,"{3841, 4074, 4047, 2633, 4045, 2635, 8}"
486,1,158,0,0.2360122,"\int (a+b \sec (c+d x))^5 \, dx","Int[(a + b*Sec[c + d*x])^5,x]","\frac{a b^2 \left(53 a^2+20 b^2\right) \tan (c+d x)}{6 d}+\frac{b \left(40 a^2 b^2+40 a^4+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^3 \left(58 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+a^5 x+\frac{11 a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}","\frac{a b^2 \left(53 a^2+20 b^2\right) \tan (c+d x)}{6 d}+\frac{b \left(40 a^2 b^2+40 a^4+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^3 \left(58 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+a^5 x+\frac{11 a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"a^5*x + (b*(40*a^4 + 40*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b^2*(53*a^2 + 20*b^2)*Tan[c + d*x])/(6*d) + (b^3*(58*a^2 + 9*b^2)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (11*a*b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b^2*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,6,12,0.5000,1,"{3782, 4056, 4048, 3770, 3767, 8}"
487,1,157,0,0.4848246,"\int \frac{\sec ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{3 b^3 d}-\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 b d}","\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{3 b^3 d}-\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"-(a*(2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^4*ArcTanh[(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)*Tan[c + d*x])/(3*b^3*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)","A",8,8,21,0.3810,1,"{3851, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
488,1,119,0,0.2752395,"\int \frac{\sec ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 a^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 a^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"((2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*Tan[c + d*x])/(b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",7,7,21,0.3333,1,"{3851, 4082, 3998, 3770, 3831, 2659, 208}"
489,1,85,0,0.1625783,"\int \frac{\sec ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\tan (c+d x)}{b d}","\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\tan (c+d x)}{b d}",1,"-((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) + (2*a^2*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Tan[c + d*x]/(b*d)","A",6,6,21,0.2857,1,"{3790, 3789, 3770, 3831, 2659, 208}"
490,1,68,0,0.1089157,"\int \frac{\sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}",1,"ArcTanh[Sin[c + d*x]]/(b*d) - (2*a*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",5,5,21,0.2381,1,"{3789, 3770, 3831, 2659, 208}"
491,1,49,0,0.0589132,"\int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{2 \tanh ^{-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 \tanh ^{-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*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",3,3,19,0.1579,1,"{3831, 2659, 208}"
492,1,59,0,0.0505581,"\int \frac{1}{a+b \sec (c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(-1),x]","\frac{x}{a}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}","\frac{x}{a}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"x/a - (2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)","A",3,3,12,0.2500,1,"{3783, 2659, 208}"
493,1,76,0,0.1015241,"\int \frac{\cos (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{2 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x}{a^2}+\frac{\sin (c+d x)}{a d}","\frac{2 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x}{a^2}+\frac{\sin (c+d x)}{a d}",1,"-((b*x)/a^2) + (2*b^2*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + Sin[c + d*x]/(a*d)","A",5,5,19,0.2632,1,"{3853, 12, 3783, 2659, 208}"
494,1,110,0,0.2777574,"\int \frac{\cos ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x]),x]","-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2+2 b^2\right)}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}","-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2+2 b^2\right)}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}",1,"((a^2 + 2*b^2)*x)/(2*a^3) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - (b*Sin[c + d*x])/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",6,6,21,0.2857,1,"{3853, 4104, 3919, 3831, 2659, 208}"
495,1,148,0,0.4585557,"\int \frac{\cos ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^3 d}+\frac{2 b^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}-\frac{b \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 a d}","\frac{\left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^3 d}+\frac{2 b^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}-\frac{b \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"-(b*(a^2 + 2*b^2)*x)/(2*a^4) + (2*b^4*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^3*d) - (b*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)","A",7,6,21,0.2857,1,"{3853, 4104, 3919, 3831, 2659, 208}"
496,1,193,0,0.6865222,"\int \frac{\cos ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + b*Sec[c + d*x]),x]","-\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^4 d}+\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(4 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}-\frac{b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}","-\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^4 d}+\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(4 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}-\frac{b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(2*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^4*d) + ((3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (b*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,6,21,0.2857,1,"{3853, 4104, 3919, 3831, 2659, 208}"
497,1,222,0,0.6124215,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","-\frac{a \left(3 a^2-2 b^2\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}","-\frac{a \left(3 a^2-2 b^2\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}",1,"((6*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(3*a^2 - 4*b^2)*ArcTanh[(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)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,8,21,0.3810,1,"{3845, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
498,1,164,0,0.3464232,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(-2*a*ArcTanh[Sin[c + d*x]])/(b^3*d) + (2*a^2*(2*a^2 - 3*b^2)*ArcTanh[(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)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,21,0.3333,1,"{3845, 4082, 3998, 3770, 3831, 2659, 208}"
499,1,117,0,0.2185817,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","-\frac{2 a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}","-\frac{2 a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"ArcTanh[Sin[c + d*x]]/(b^2*d) - (2*a*(a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,21,0.2857,1,"{3839, 3998, 3770, 3831, 2659, 208}"
500,1,85,0,0.1269982,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{a \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}","\frac{a \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3836, 12, 3831, 2659, 208}"
501,1,86,0,0.1030982,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}",1,"(2*a*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,19,0.2632,1,"{3833, 12, 3831, 2659, 208}"
502,1,109,0,0.1687309,"\int \frac{1}{(a+b \sec (c+d x))^2} \, dx","Int[(a + b*Sec[c + d*x])^(-2),x]","-\frac{2 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x}{a^2}","-\frac{2 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x}{a^2}",1,"x/a^2 - (2*b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,12,0.4167,1,"{3785, 3919, 3831, 2659, 208}"
503,1,146,0,0.3281201,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 b x}{a^3}","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 b x}{a^3}",1,"(-2*b*x)/a^3 + (2*b^2*(3*a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2 - 2*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,19,0.3158,1,"{3847, 4104, 3919, 3831, 2659, 208}"
504,1,208,0,0.5854896,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","-\frac{b \left(2 a^2-3 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2+6 b^2\right)}{2 a^4}","-\frac{b \left(2 a^2-3 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2+6 b^2\right)}{2 a^4}",1,"((a^2 + 6*b^2)*x)/(2*a^4) - (2*b^3*(4*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(2*a^2 - 3*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,6,21,0.2857,1,"{3847, 4104, 3919, 3831, 2659, 208}"
505,1,261,0,0.8354029,"\int \frac{\cos ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{\left(7 a^2 b^2+2 a^4-12 b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\frac{\left(a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{2 b^4 \left(5 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x) \cos ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b x \left(a^2+4 b^2\right)}{a^5}","\frac{\left(7 a^2 b^2+2 a^4-12 b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\frac{\left(a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{2 b^4 \left(5 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x) \cos ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b x \left(a^2+4 b^2\right)}{a^5}",1,"-((b*(a^2 + 4*b^2)*x)/a^5) + (2*b^4*(5*a^2 - 4*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,6,21,0.2857,1,"{3847, 4104, 3919, 3831, 2659, 208}"
506,1,230,0,0.7074574,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{\left(3 a^2-2 b^2\right) \tan (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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{3 a^3 \left(a^2-2 b^2\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}","\frac{\left(3 a^2-2 b^2\right) \tan (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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{3 a^3 \left(a^2-2 b^2\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(-3*a*ArcTanh[Sin[c + d*x]])/(b^4*d) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTanh[(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)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (3*a^3*(a^2 - 2*b^2)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,21,0.3810,1,"{3845, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
507,1,188,0,0.4099816,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(2 a^2-5 b^2\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^3 d}","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \left(2 a^2-5 b^2\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"ArcTanh[Sin[c + d*x]]/(b^3*d) - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(2*a^2 - 5*b^2)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,21,0.3333,1,"{3845, 4080, 3998, 3770, 3831, 2659, 208}"
508,1,149,0,0.231627,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}",1,"((a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2 - 4*b^2)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,21,0.2857,1,"{3839, 4003, 12, 3831, 2659, 208}"
509,1,134,0,0.190906,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{\left(a^2+2 b^2\right) \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}","\frac{\left(a^2+2 b^2\right) \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"(-3*a*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) + (a*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2 + 2*b^2)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,21,0.2857,1,"{3836, 4003, 12, 3831, 2659, 208}"
510,1,133,0,0.1754489,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^3,x]","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a b \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a b \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}",1,"((2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (3*a*b*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,19,0.3158,1,"{3833, 4003, 12, 3831, 2659, 208}"
511,1,173,0,0.3098369,"\int \frac{1}{(a+b \sec (c+d x))^3} \, dx","Int[(a + b*Sec[c + d*x])^(-3),x]","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(5 a^2-2 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x}{a^3}","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(5 a^2-2 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x}{a^3}",1,"x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,12,0.5000,1,"{3785, 4060, 3919, 3831, 2659, 208}"
512,1,223,0,0.6234476,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x])^3,x]","\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{3 b^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 b x}{a^4}","\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{3 b^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 b x}{a^4}",1,"(-3*b*x)/a^4 + (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (3*b^2*(2*a^2 - b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,19,0.3684,1,"{3847, 4100, 4104, 3919, 3831, 2659, 208}"
513,1,296,0,0.9981535,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","-\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(7 a^2-4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x \left(a^2+12 b^2\right)}{2 a^5}","-\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left(7 a^2-4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x \left(a^2+12 b^2\right)}{2 a^5}",1,"((a^2 + 12*b^2)*x)/(2*a^5) - (b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(7*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,7,21,0.3333,1,"{3847, 4100, 4104, 3919, 3831, 2659, 208}"
514,1,316,0,1.1061605,"\int \frac{\sec ^6(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^6/(a + b*Sec[c + d*x])^4,x]","\frac{\left(-23 a^2 b^2+12 a^4+6 b^4\right) \tan (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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(4 a^2-9 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a^3 \left(-11 a^2 b^2+4 a^4+12 b^4\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{4 a \tanh ^{-1}(\sin (c+d x))}{b^5 d}","\frac{\left(-23 a^2 b^2+12 a^4+6 b^4\right) \tan (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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(4 a^2-9 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a^3 \left(-11 a^2 b^2+4 a^4+12 b^4\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{4 a \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"(-4*a*ArcTanh[Sin[c + d*x]])/(b^5*d) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTanh[(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)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,9,21,0.4286,1,"{3845, 4098, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
515,1,259,0,0.7517951,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a^3 \left(3 a^2-8 b^2\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \left(-28 a^2 b^2+9 a^4+34 b^4\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^4 d}","-\frac{a \left(-7 a^4 b^2+8 a^2 b^4+2 a^6-8 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a^3 \left(3 a^2-8 b^2\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \left(-28 a^2 b^2+9 a^4+34 b^4\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"ArcTanh[Sin[c + d*x]]/(b^4*d) - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a^3*(3*a^2 - 8*b^2)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,8,21,0.3810,1,"{3845, 4090, 4080, 3998, 3770, 3831, 2659, 208}"
516,1,222,0,0.4212823,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^4,x]","-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(-5 a^2 b^2+2 a^4+18 b^4\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}","-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(-5 a^2 b^2+2 a^4+18 b^4\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"-((b*(3*a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a^2*(2*a^2 - 7*b^2)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,7,21,0.3333,1,"{3845, 4080, 4003, 12, 3831, 2659, 208}"
517,1,206,0,0.3534129,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^4,x]","\frac{a \left(a^2+4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(a^2-6 b^2\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(-10 a^2 b^2+a^4-6 b^4\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}","\frac{a \left(a^2+4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(a^2-6 b^2\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(-10 a^2 b^2+a^4-6 b^4\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"(a*(a^2 + 4*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2 - 6*b^2)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,21,0.2857,1,"{3839, 4003, 12, 3831, 2659, 208}"
518,1,192,0,0.3073181,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^4,x]","-\frac{b \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}","-\frac{b \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}",1,"-((b*(4*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^2 + 13*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,21,0.2857,1,"{3836, 4003, 12, 3831, 2659, 208}"
519,1,184,0,0.3072892,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^4,x]","\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{5 a b \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}","\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{5 a b \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}",1,"(a*(2*a^2 + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (5*a*b*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,19,0.3158,1,"{3833, 4003, 12, 3831, 2659, 208}"
520,1,242,0,0.5352638,"\int \frac{1}{(a+b \sec (c+d x))^4} \, dx","Int[(a + b*Sec[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) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-17 a^2 b^2+26 a^4+6 b^4\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{b^2 \left(8 a^2-3 b^2\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x}{a^4}","-\frac{b \left(-8 a^4 b^2+7 a^2 b^4+8 a^6-2 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-17 a^2 b^2+26 a^4+6 b^4\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{b^2 \left(8 a^2-3 b^2\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x}{a^4}",1,"x/a^4 - (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2 - 3*b^2)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,12,0.5000,1,"{3785, 4060, 3919, 3831, 2659, 208}"
521,1,299,0,1.0370429,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x])^4,x]","\frac{\left(-65 a^4 b^2+68 a^2 b^4+6 a^6-24 b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(-35 a^4 b^2+28 a^2 b^4+20 a^6-8 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-11 a^2 b^2+12 a^4+4 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{b^2 \left(9 a^2-4 b^2\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{4 b x}{a^5}","\frac{\left(-65 a^4 b^2+68 a^2 b^4+6 a^6-24 b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(-35 a^4 b^2+28 a^2 b^4+20 a^6-8 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-11 a^2 b^2+12 a^4+4 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{b^2 \left(9 a^2-4 b^2\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{4 b x}{a^5}",1,"(-4*b*x)/a^5 + (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(9*a^2 - 4*b^2)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,7,19,0.3684,1,"{3847, 4100, 4104, 3919, 3831, 2659, 208}"
522,1,387,0,1.4559742,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^4,x]","-\frac{b \left(-146 a^4 b^2+167 a^2 b^4+24 a^6-60 b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(-23 a^4 b^2+27 a^2 b^4+a^6-10 b^6\right) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}-\frac{b^3 \left(-84 a^4 b^2+69 a^2 b^4+40 a^6-20 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-53 a^2 b^2+48 a^4+20 b^4\right) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{5 b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x \left(a^2+20 b^2\right)}{2 a^6}","-\frac{b \left(-146 a^4 b^2+167 a^2 b^4+24 a^6-60 b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(-23 a^4 b^2+27 a^2 b^4+a^6-10 b^6\right) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}-\frac{b^3 \left(-84 a^4 b^2+69 a^2 b^4+40 a^6-20 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(-53 a^2 b^2+48 a^4+20 b^4\right) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{5 b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x \left(a^2+20 b^2\right)}{2 a^6}",1,"((a^2 + 20*b^2)*x)/(2*a^6) - (b^3*(40*a^6 - 84*a^4*b^2 + 69*a^2*b^4 - 20*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(24*a^6 - 146*a^4*b^2 + 167*a^2*b^4 - 60*b^6)*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6 - 23*a^4*b^2 + 27*a^2*b^4 - 10*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (5*b^2*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(48*a^4 - 53*a^2*b^2 + 20*b^4)*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,7,21,0.3333,1,"{3847, 4100, 4104, 3919, 3831, 2659, 208}"
523,1,31,0,0.0308961,"\int \frac{1}{3+5 \sec (c+d x)} \, dx","Int[(3 + 5*Sec[c + d*x])^(-1),x]","\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{6 d}-\frac{x}{12}","\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{6 d}-\frac{x}{12}",1,"-x/12 + (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(6*d)","A",2,2,12,0.1667,1,"{3783, 2657}"
524,1,56,0,0.0796837,"\int \frac{1}{(3+5 \sec (c+d x))^2} \, dx","Int[(3 + 5*Sec[c + d*x])^(-2),x]","-\frac{25 \tan (c+d x)}{48 d (5 \sec (c+d x)+3)}+\frac{35 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{288 d}+\frac{29 x}{576}","-\frac{25 \tan (c+d x)}{48 d (5 \sec (c+d x)+3)}+\frac{35 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{288 d}+\frac{29 x}{576}",1,"(29*x)/576 + (35*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(288*d) - (25*Tan[c + d*x])/(48*d*(3 + 5*Sec[c + d*x]))","A",4,4,12,0.3333,1,"{3785, 3919, 3831, 2657}"
525,1,81,0,0.116301,"\int \frac{1}{(3+5 \sec (c+d x))^3} \, dx","Int[(3 + 5*Sec[c + d*x])^(-3),x]","-\frac{125 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{96 d (5 \sec (c+d x)+3)^2}+\frac{3055 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{27648 d}-\frac{1007 x}{55296}","-\frac{125 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{96 d (5 \sec (c+d x)+3)^2}+\frac{3055 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{27648 d}-\frac{1007 x}{55296}",1,"(-1007*x)/55296 + (3055*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(27648*d) - (25*Tan[c + d*x])/(96*d*(3 + 5*Sec[c + d*x])^2) - (125*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x]))","A",5,5,12,0.4167,1,"{3785, 4060, 3919, 3831, 2657}"
526,1,106,0,0.1579041,"\int \frac{1}{(3+5 \sec (c+d x))^4} \, dx","Int[(3 + 5*Sec[c + d*x])^(-4),x]","-\frac{16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac{25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac{11215 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{1327104 d}+\frac{21553 x}{2654208}","-\frac{16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac{25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac{11215 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{1327104 d}+\frac{21553 x}{2654208}",1,"(21553*x)/2654208 + (11215*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1327104*d) - (25*Tan[c + d*x])/(144*d*(3 + 5*Sec[c + d*x])^3) - (25*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x])^2) - (16925*Tan[c + d*x])/(221184*d*(3 + 5*Sec[c + d*x]))","A",6,5,12,0.4167,1,"{3785, 4060, 3919, 3831, 2657}"
527,1,70,0,0.0349103,"\int \frac{1}{5+3 \sec (c+d x)} \, dx","Int[(5 + 3*Sec[c + d*x])^(-1),x]","\frac{3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}+\frac{x}{5}","\frac{3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}+\frac{x}{5}",1,"x/5 + (3*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(20*d) - (3*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(20*d)","A",3,3,12,0.2500,1,"{3783, 2659, 206}"
528,1,95,0,0.0926357,"\int \frac{1}{(5+3 \sec (c+d x))^2} \, dx","Int[(5 + 3*Sec[c + d*x])^(-2),x]","\frac{9 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}+\frac{123 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}-\frac{123 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}+\frac{x}{25}","\frac{9 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}+\frac{123 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}-\frac{123 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}+\frac{x}{25}",1,"x/25 + (123*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(1600*d) - (123*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(1600*d) + (9*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x]))","A",5,5,12,0.4167,1,"{3785, 3919, 3831, 2659, 206}"
529,1,120,0,0.1330313,"\int \frac{1}{(5+3 \sec (c+d x))^3} \, dx","Int[(5 + 3*Sec[c + d*x])^(-3),x]","\frac{963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac{9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac{8361 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}-\frac{8361 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}+\frac{x}{125}","\frac{963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac{9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac{8361 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}-\frac{8361 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}+\frac{x}{125}",1,"x/125 + (8361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(256000*d) - (8361*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(256000*d) + (9*Tan[c + d*x])/(160*d*(5 + 3*Sec[c + d*x])^2) + (963*Tan[c + d*x])/(12800*d*(5 + 3*Sec[c + d*x]))","A",6,6,12,0.5000,1,"{3785, 4060, 3919, 3831, 2659, 206}"
530,1,145,0,0.1803818,"\int \frac{1}{(5+3 \sec (c+d x))^4} \, dx","Int[(5 + 3*Sec[c + d*x])^(-4),x]","\frac{38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac{519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac{3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac{278151 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}-\frac{278151 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}+\frac{x}{625}","\frac{38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac{519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac{3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac{278151 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}-\frac{278151 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}+\frac{x}{625}",1,"x/625 + (278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(20480000*d) - (278151*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(20480000*d) + (3*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])^3) + (519*Tan[c + d*x])/(12800*d*(5 + 3*Sec[c + d*x])^2) + (38733*Tan[c + d*x])/(1024000*d*(5 + 3*Sec[c + d*x]))","A",7,6,12,0.5000,1,"{3785, 4060, 3919, 3831, 2659, 206}"
531,1,292,0,0.4415075,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 (a-b) \sqrt{a+b} \left(2 a^2-9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (a-b) \sqrt{a+b} (2 a+9 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}-\frac{4 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}","\frac{2 (a-b) \sqrt{a+b} \left(2 a^2-9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (a-b) \sqrt{a+b} (2 a+9 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}-\frac{4 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}",1,"(2*(a - b)*Sqrt[a + b]*(2*a^2 - 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(2*a + 9*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) - (4*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)","A",5,5,23,0.2174,1,"{3840, 4002, 4005, 3832, 4004}"
532,1,241,0,0.2776325,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}",1,"(-2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) - (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",4,4,23,0.1739,1,"{3835, 4005, 3832, 4004}"
533,1,209,0,0.1634588,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)","A",3,3,21,0.1429,1,"{3829, 3832, 4004}"
534,1,125,0,0.0288398,"\int \sqrt{a+b \sec (c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(-2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d)","A",1,1,14,0.07143,1,"{3780}"
535,1,330,0,0.3230311,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,21,0.2857,1,"{3857, 4059, 3921, 3784, 3832, 4004}"
536,1,396,0,0.5987464,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \left(4 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}","-\frac{\sqrt{a+b} \left(4 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}",1,"((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2 - b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,23,0.3043,1,"{3857, 4104, 4058, 3921, 3784, 3832, 4004}"
537,1,405,0,0.8415047,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(8 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b+8 a^3+39 a b^2-147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(33 a^2 b^2+8 a^4+147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}","\frac{2 \left(8 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 b+8 a^3+39 a b^2-147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(33 a^2 b^2+8 a^4+147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(-2*(a - b)*Sqrt[a + b]*(8*a^4 + 33*a^2*b^2 + 147*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^3 + 6*a^2*b + 39*a*b^2 - 147*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*a*(8*a^2 + 39*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) - (8*a*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)","A",7,6,23,0.2609,1,"{3865, 4082, 4002, 4005, 3832, 4004}"
538,1,342,0,0.5987723,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 \left(6 a^2-25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b d}+\frac{2 (a-b) \sqrt{a+b} \left(6 a^2+57 a b-25 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \left(3 a^2-41 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}","-\frac{2 \left(6 a^2-25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b d}+\frac{2 (a-b) \sqrt{a+b} \left(6 a^2+57 a b-25 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \left(3 a^2-41 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}",1,"(4*a*(a - b)*Sqrt[a + b]*(3*a^2 - 41*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2 + 57*a*b - 25*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) - (2*(6*a^2 - 25*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) - (4*a*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)","A",6,5,23,0.2174,1,"{3840, 4002, 4005, 3832, 4004}"
539,1,282,0,0.4098321,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 (a-b) \sqrt{a+b} \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{2 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}-\frac{2 (a-3 b) (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b d}","-\frac{2 (a-b) \sqrt{a+b} \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{2 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}-\frac{2 (a-3 b) (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b d}",1,"(-2*(a - b)*Sqrt[a + b]*(a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^2*d) - (2*(a - 3*b)*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b*d) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",5,5,23,0.2174,1,"{3835, 4002, 4005, 3832, 4004}"
540,1,249,0,0.2916528,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 (a-b) (3 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{8 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}","\frac{2 b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 (a-b) (3 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{8 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}",1,"(-8*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*(a - b)*(3*a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{3830, 4005, 3832, 4004}"
541,1,309,0,0.2198079,"\int (a+b \sec (c+d x))^{3/2} \, dx","Int[(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 (2 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 a \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}","\frac{2 (2 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 a \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(2*a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",5,5,14,0.3571,1,"{3781, 3921, 3784, 3832, 4004}"
542,1,334,0,0.3341628,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}","\frac{a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (3*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,21,0.2857,1,"{3864, 4058, 3921, 3784, 3832, 4004}"
543,1,390,0,0.5438146,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2),x]","-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{5 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+5 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}","-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{5 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+5 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}",1,"(5*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a + 5*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2 + 3*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (5*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,23,0.3043,1,"{3864, 4104, 4058, 3921, 3784, 3832, 4004}"
544,1,463,0,1.0407783,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(8 a^2+81 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(57 a^2 b^2+8 a^4+135 b^4\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{693 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(57 a^2 b^2+6 a^3 b+8 a^4-606 a b^3+135 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^3 d}-\frac{2 a (a-b) \sqrt{a+b} \left(51 a^2 b^2+8 a^4+741 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^4 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}","\frac{2 \left(8 a^2+81 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(57 a^2 b^2+8 a^4+135 b^4\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{693 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(57 a^2 b^2+6 a^3 b+8 a^4-606 a b^3+135 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^3 d}-\frac{2 a (a-b) \sqrt{a+b} \left(51 a^2 b^2+8 a^4+741 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^4 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(-2*a*(a - b)*Sqrt[a + b]*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^4 + 6*a^3*b + 57*a^2*b^2 - 606*a*b^3 + 135*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^3*d) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) - (8*a*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)","A",8,6,23,0.2609,1,"{3865, 4082, 4002, 4005, 3832, 4004}"
545,1,399,0,0.7796781,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \left(10 a^2-49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left(165 a^2 b+10 a^3-114 a b^2+147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(-279 a^2 b^2+10 a^4-147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}","-\frac{2 \left(10 a^2-49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left(165 a^2 b+10 a^3-114 a b^2+147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(-279 a^2 b^2+10 a^4-147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}",1,"(2*(a - b)*Sqrt[a + b]*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3 + 165*a^2*b - 114*a*b^2 + 147*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)","A",7,5,23,0.2174,1,"{3840, 4002, 4005, 3832, 4004}"
546,1,333,0,0.5655564,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(3 a^2+5 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}-\frac{2 (a-b) \sqrt{a+b} \left(3 a^2-24 a b+5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b d}-\frac{2 a (a-b) \sqrt{a+b} \left(3 a^2+29 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac{2 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}","\frac{2 \left(3 a^2+5 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}-\frac{2 (a-b) \sqrt{a+b} \left(3 a^2-24 a b+5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b d}-\frac{2 a (a-b) \sqrt{a+b} \left(3 a^2+29 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac{2 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}",1,"(-2*a*(a - b)*Sqrt[a + b]*(3*a^2 + 29*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b^2*d) - (2*(a - b)*Sqrt[a + b]*(3*a^2 - 24*a*b + 5*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b*d) + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",6,5,23,0.2174,1,"{3835, 4002, 4005, 3832, 4004}"
547,1,296,0,0.4504732,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 (a-b) \sqrt{a+b} \left(15 a^2-8 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{2 b \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{16 a b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}","\frac{2 (a-b) \sqrt{a+b} \left(15 a^2-8 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{2 b \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{16 a b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}",1,"(-2*(a - b)*Sqrt[a + b]*(23*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(a - b)*Sqrt[a + b]*(15*a^2 - 8*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (16*a*b*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",5,5,21,0.2381,1,"{3830, 4002, 4005, 3832, 4004}"
548,1,352,0,0.3341144,"\int (a+b \sec (c+d x))^{5/2} \, dx","Int[(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \sqrt{a+b} \left(9 a^2-7 a b+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}-\frac{2 a^2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{14 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}","\frac{2 \sqrt{a+b} \left(9 a^2-7 a b+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}-\frac{2 a^2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{14 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}",1,"(-14*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) + (2*Sqrt[a + b]*(9*a^2 - 7*a*b + b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a^2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,14,0.4286,1,"{3782, 4058, 3921, 3784, 3832, 4004}"
549,1,353,0,0.3445357,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{a+b} \left(a^2+6 a b-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \left(a^2-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}-\frac{5 a b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}","\frac{\sqrt{a+b} \left(a^2+6 a b-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \left(a^2-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}-\frac{5 a b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"((a - b)*Sqrt[a + b]*(a^2 - 2*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(a^2 + 6*a*b - 2*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (5*a*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,21,0.2857,1,"{3841, 4058, 3921, 3784, 3832, 4004}"
550,1,399,0,0.6320099,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{a+b} \left(2 a^2+9 a b+8 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{9 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}","\frac{\sqrt{a+b} \left(2 a^2+9 a b+8 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{9 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}",1,"(9*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a^2 + 9*a*b + 8*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2 + 15*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (9*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,23,0.3043,1,"{3841, 4104, 4058, 3921, 3784, 3832, 4004}"
551,1,460,0,0.9333371,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\left(16 a^2+33 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(16 a^2+26 a b+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{5 b \sqrt{a+b} \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}","\frac{\left(16 a^2+33 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(16 a^2+26 a b+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{5 b \sqrt{a+b} \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}",1,"((a - b)*Sqrt[a + b]*(16*a^2 + 33*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2 + 26*a*b + 33*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (5*b*Sqrt[a + b]*(4*a^2 + b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2 + 33*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (13*a*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a^2*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,7,23,0.3043,1,"{3841, 4104, 4058, 3921, 3784, 3832, 4004}"
552,1,530,0,1.3001266,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]","\frac{b \left(284 a^2+15 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left(36 a^2+59 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\sqrt{a+b} \left(284 a^2 b+72 a^3+118 a b^2+15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{(a-b) \sqrt{a+b} \left(284 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}-\frac{\sqrt{a+b} \left(120 a^2 b^2+48 a^4-5 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}","\frac{b \left(284 a^2+15 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left(36 a^2+59 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\sqrt{a+b} \left(284 a^2 b+72 a^3+118 a b^2+15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{(a-b) \sqrt{a+b} \left(284 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}-\frac{\sqrt{a+b} \left(120 a^2 b^2+48 a^4-5 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}",1,"((a - b)*Sqrt[a + b]*(284*a^2 + 15*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(72*a^3 + 284*a^2*b + 118*a*b^2 + 15*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4 + 120*a^2*b^2 - 5*b^4)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (b*(284*a^2 + 15*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2 + 59*b^2)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (17*a*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",9,7,23,0.3043,1,"{3841, 4104, 4058, 3921, 3784, 3832, 4004}"
553,1,403,0,0.4994694,"\int (a+b \sec (c+d x))^{7/2} \, dx","Int[(a + b*Sec[c + d*x])^(7/2),x]","\frac{2 \sqrt{a+b} \left(-58 a^2 b+60 a^3+22 a b^2-9 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}-\frac{2 (a-b) \sqrt{a+b} \left(58 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}-\frac{2 a^3 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{26 a b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b^2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}","\frac{2 \sqrt{a+b} \left(-58 a^2 b+60 a^3+22 a b^2-9 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}-\frac{2 (a-b) \sqrt{a+b} \left(58 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}-\frac{2 a^3 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{26 a b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b^2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(-2*(a - b)*Sqrt[a + b]*(58*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) + (2*Sqrt[a + b]*(60*a^3 - 58*a^2*b + 22*a*b^2 - 9*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*a^3*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (26*a*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b^2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,7,14,0.5000,1,"{3782, 4056, 4058, 3921, 3784, 3832, 4004}"
554,1,359,0,0.6725182,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \left(24 a^2+25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}+\frac{2 \sqrt{a+b} \left(-12 a^2 b+48 a^3+44 a b^2+25 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{8 a (a-b) \sqrt{a+b} \left(12 a^2+11 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}-\frac{12 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}","\frac{2 \left(24 a^2+25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}+\frac{2 \sqrt{a+b} \left(-12 a^2 b+48 a^3+44 a b^2+25 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{8 a (a-b) \sqrt{a+b} \left(12 a^2+11 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}-\frac{12 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}",1,"(8*a*(a - b)*Sqrt[a + b]*(12*a^2 + 11*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + b]*(48*a^3 - 12*a^2*b + 44*a*b^2 + 25*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(24*a^2 + 25*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) - (12*a*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)","A",6,6,23,0.2609,1,"{3860, 4092, 4082, 4005, 3832, 4004}"
555,1,301,0,0.4230954,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \sqrt{a+b} \left(8 a^2-2 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}","-\frac{2 \sqrt{a+b} \left(8 a^2-2 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"(-2*(a - b)*Sqrt[a + b]*(8*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(8*a^2 - 2*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (8*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",5,5,23,0.2174,1,"{3860, 4082, 4005, 3832, 4004}"
556,1,244,0,0.2749162,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}","\frac{2 \sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(4*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)","A",4,4,23,0.1739,1,"{3840, 4005, 3832, 4004}"
557,1,204,0,0.1560825,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)","A",3,3,23,0.1304,1,"{3837, 3832, 4004}"
558,1,99,0,0.0397408,"\int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)","A",1,1,21,0.04762,1,"{3832}"
559,1,106,0,0.0216419,"\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[1/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",1,1,14,0.07143,1,"{3784}"
560,1,338,0,0.2654678,"\int \frac{\cos (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}","\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",6,6,21,0.2857,1,"{3861, 4059, 3921, 3784, 3832, 4004}"
561,1,401,0,0.5098536,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}-\frac{3 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}+\frac{(2 a-3 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{3 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}","-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}-\frac{3 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}+\frac{(2 a-3 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{3 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"(-3*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((2*a - 3*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(4*a^2 + 3*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - (3*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)","A",7,7,23,0.3043,1,"{3863, 4104, 4058, 3921, 3784, 3832, 4004}"
562,1,399,0,0.7747441,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{2 (4 a+3 b) \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^4 d \sqrt{a+b}}-\frac{2 \left(-8 a^2 b^2+16 a^4-3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^5 d \sqrt{a+b}}","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{2 (4 a+3 b) \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^4 d \sqrt{a+b}}-\frac{2 \left(-8 a^2 b^2+16 a^4-3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^5 d \sqrt{a+b}}",1,"(-2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^5*Sqrt[a + b]*d) - (2*(4*a + 3*b)*(4*a^2 + b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^4*Sqrt[a + b]*d) - (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a*(8*a^2 - 3*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",6,6,23,0.2609,1,"{3845, 4092, 4082, 4005, 3832, 4004}"
563,1,325,0,0.5035773,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 a \left(8 a^2-5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}+\frac{2 (2 a+b) (4 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}","-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 a \left(8 a^2-5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}+\frac{2 (2 a+b) (4 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}",1,"(2*a*(8*a^2 - 5*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(2*a + b)*(4*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d)","A",5,5,23,0.2174,1,"{3845, 4082, 4005, 3832, 4004}"
564,1,257,0,0.3194855,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(2 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}-\frac{2 (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}","-\frac{2 a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(2 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}-\frac{2 (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"(-2*(2*a^2 - b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) - (2*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*a^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,23,0.1739,1,"{3839, 4005, 3832, 4004}"
565,1,237,0,0.2929734,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 a \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}","\frac{2 a \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}",1,"(2*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*a*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,23,0.1739,1,"{3836, 4005, 3832, 4004}"
566,1,236,0,0.2285105,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 b \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}","-\frac{2 b \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}",1,"(-2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*b*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,21,0.2381,1,"{3833, 21, 3829, 3832, 4004}"
567,1,347,0,0.3163005,"\int \frac{1}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(a + b*Sec[c + d*x])^(-3/2),x]","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,14,0.4286,1,"{3785, 4058, 3921, 3784, 3832, 4004}"
568,1,396,0,0.4975148,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{b \left(a^2-3 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{(a+3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}","\frac{b \left(a^2-3 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{(a+3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((a + 3*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (3*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + Sin[c + d*x]/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2 - 3*b^2)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,21,0.3333,1,"{3846, 4061, 4058, 3921, 3784, 3832, 4004}"
569,1,470,0,0.7752197,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{b^2 \left(7 a^2-15 b^2\right) \tan (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-5 a b-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}-\frac{\left(7 a^2-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^4 d}-\frac{5 b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}","-\frac{b^2 \left(7 a^2-15 b^2\right) \tan (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-5 a b-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}-\frac{\left(7 a^2-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^4 d}-\frac{5 b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"-((7*a^2 - 15*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) + ((2*a^2 - 5*a*b - 15*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(4*a^2 + 15*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - (5*b*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) - (b^2*(7*a^2 - 15*b^2)*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,23,0.3478,1,"{3846, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
570,1,427,0,0.9327914,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{4 a^3 \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-16 a^2 b^2+12 a^3 b+16 a^4-9 a b^3-b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}+\frac{8 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d (a-b) (a+b)^{3/2}}","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{4 a^3 \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-16 a^2 b^2+12 a^3 b+16 a^4-9 a b^3-b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}+\frac{8 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d (a-b) (a+b)^{3/2}}",1,"(8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) + (2*(16*a^4 + 12*a^3*b - 16*a^2*b^2 - 9*a*b^3 - b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(2*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",6,6,23,0.2609,1,"{3845, 4090, 4082, 4005, 3832, 4004}"
571,1,362,0,0.5912054,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{8 a^2 \left(a^2-2 b^2\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(6 a^2 b+8 a^3-9 a b^2-3 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}-\frac{2 \left(-15 a^2 b^2+8 a^4+3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}","-\frac{8 a^2 \left(a^2-2 b^2\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(6 a^2 b+8 a^3-9 a b^2-3 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}-\frac{2 \left(-15 a^2 b^2+8 a^4+3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}",1,"(-2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*(8*a^3 + 6*a^2*b - 9*a*b^2 - 3*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (8*a^2*(a^2 - 2*b^2)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3845, 4080, 4005, 3832, 4004}"
572,1,337,0,0.5015033,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^2+3 a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}","-\frac{2 a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^2+3 a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"(4*a*(a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2 + 3*a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a*(a^2 - 3*b^2)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3839, 4003, 4005, 3832, 4004}"
573,1,317,0,0.4364191,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(a^2+3 b^2\right) \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 (a-3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}","\frac{2 \left(a^2+3 b^2\right) \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 (a-3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}",1,"(2*(a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a - 3*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*a*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2 + 3*b^2)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,23,0.2174,1,"{3836, 4003, 4005, 3832, 4004}"
574,1,304,0,0.4079946,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{8 a b \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 (3 a-b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}-\frac{8 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}","-\frac{8 a b \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 (3 a-b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}-\frac{8 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}",1,"(-8*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*(3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*b*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (8*a*b*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,21,0.2381,1,"{3833, 4003, 4005, 3832, 4004}"
575,1,448,0,0.5637366,"\int \frac{1}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(a + b*Sec[c + d*x])^(-5/2),x]","\frac{2 b^2 \left(7 a^2-3 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(6 a^2-a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(7 a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}","\frac{2 b^2 \left(7 a^2-3 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(6 a^2-a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(7 a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}",1,"(2*(7*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*(6*a^2 - a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,14,0.5000,1,"{3785, 4060, 4058, 3921, 3784, 3832, 4004}"
576,1,510,0,0.8141558,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + b*Sec[c + d*x])^(5/2),x]","\frac{b \left(-26 a^2 b^2+3 a^4+15 b^4\right) \tan (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{b \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\left(21 a^2 b+3 a^3-5 a b^2-15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}+\frac{\left(-26 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{5 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{\sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}","\frac{b \left(-26 a^2 b^2+3 a^4+15 b^4\right) \tan (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{b \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\left(21 a^2 b+3 a^3-5 a b^2-15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}+\frac{\left(-26 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{5 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{\sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"((3*a^4 - 26*a^2*b^2 + 15*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + ((3*a^3 + 21*a^2*b - 5*a*b^2 - 15*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (5*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + Sin[c + d*x]/(a*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^2 - 5*b^2)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,21,0.3810,1,"{3846, 4061, 4060, 4058, 3921, 3784, 3832, 4004}"
577,1,562,0,1.1666495,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{b^2 \left(-170 a^2 b^2+33 a^4+105 b^4\right) \tan (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{b^2 \left(27 a^2-35 b^2\right) \tan (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{(a+3 b) \left(-45 a^2 b+6 a^3+35 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac{\left(-170 a^2 b^2+33 a^4+105 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} \left(4 a^2+35 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^5 d}-\frac{7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}","-\frac{b^2 \left(-170 a^2 b^2+33 a^4+105 b^4\right) \tan (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{b^2 \left(27 a^2-35 b^2\right) \tan (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{(a+3 b) \left(-45 a^2 b+6 a^3+35 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac{\left(-170 a^2 b^2+33 a^4+105 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} \left(4 a^2+35 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^5 d}-\frac{7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"-((33*a^4 - 170*a^2*b^2 + 105*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*(a + b)^(3/2)*d) + ((a + 3*b)*(6*a^3 - 45*a^2*b + 35*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*(a + b)^(3/2)*d) - (Sqrt[a + b]*(4*a^2 + 35*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - (7*b*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(27*a^2 - 35*b^2)*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(33*a^4 - 170*a^2*b^2 + 105*b^4)*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,23,0.3478,1,"{3846, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
578,1,535,0,0.8600805,"\int \frac{1}{(a+b \sec (c+d x))^{7/2}} \, dx","Int[(a + b*Sec[c + d*x])^(-7/2),x]","\frac{2 b^2 \left(-41 a^2 b^2+58 a^4+15 b^4\right) \tan (c+d x)}{15 a^3 d \left(a^2-b^2\right)^3 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \left(13 a^2-5 b^2\right) \tan (c+d x)}{15 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \tan (c+d x)}{5 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{5/2}}-\frac{2 \left(-36 a^2 b^2-13 a^3 b+45 a^4+5 a b^3+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}+\frac{2 \left(-41 a^2 b^2+58 a^4+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}","\frac{2 b^2 \left(-41 a^2 b^2+58 a^4+15 b^4\right) \tan (c+d x)}{15 a^3 d \left(a^2-b^2\right)^3 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \left(13 a^2-5 b^2\right) \tan (c+d x)}{15 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \tan (c+d x)}{5 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{5/2}}-\frac{2 \left(-36 a^2 b^2-13 a^3 b+45 a^4+5 a b^3+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}+\frac{2 \left(-41 a^2 b^2+58 a^4+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}",1,"(2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*(45*a^4 - 13*a^3*b - 36*a^2*b^2 + 5*a*b^3 + 15*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (2*b^2*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) + (2*b^2*(13*a^2 - 5*b^2)*Tan[c + d*x])/(15*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(15*a^3*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])","A",8,7,14,0.5000,1,"{3785, 4060, 4058, 3921, 3784, 3832, 4004}"
579,1,151,0,0.099222,"\int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x]),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) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}","\frac{2 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) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",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) + (6*b*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*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,5,21,0.2381,1,"{3787, 3768, 3771, 2641, 2639}"
580,1,123,0,0.0911815,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x]),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 \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) \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(-2*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*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,5,21,0.2381,1,"{3787, 3768, 3771, 2639, 2641}"
581,1,97,0,0.0754024,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]),x]","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",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 + (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,5,21,0.2381,1,"{3787, 3771, 2641, 3768, 2639}"
582,1,75,0,0.062732,"\int \frac{a+b \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[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 \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 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","A",5,4,21,0.1905,1,"{3787, 3771, 2639, 2641}"
583,1,101,0,0.0772574,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[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 \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 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*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,21,0.2381,1,"{3787, 3769, 3771, 2641, 2639}"
584,1,127,0,0.0892598,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[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)}{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 \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)}{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,"(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) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,5,21,0.2381,1,"{3787, 3769, 3771, 2639, 2641}"
585,1,151,0,0.0981735,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])/Sec[c + d*x]^(7/2),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{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)}{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{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) + (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*b*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",8,5,21,0.2381,1,"{3787, 3769, 3771, 2641, 2639}"
586,1,200,0,0.1483291,"\int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\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) \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 b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}","\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\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) \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 b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",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) + (12*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*a^2 + 5*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*b^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,6,23,0.2609,1,"{3788, 3768, 3771, 2639, 4046, 2641}"
587,1,175,0,0.1290388,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\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) \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 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}","\frac{2 \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\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) \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 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",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*(5*a^2 + 3*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*b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,6,23,0.2609,1,"{3788, 3768, 3771, 2641, 4046, 2639}"
588,1,135,0,0.1096503,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^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 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 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}","\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 \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 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",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) + (4*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,23,0.2609,1,"{3788, 3768, 3771, 2639, 4046, 2641}"
589,1,108,0,0.1070812,"\int \frac{(a+b \sec (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^2/Sqrt[Sec[c + d*x]],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{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 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{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{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 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{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*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",6,5,23,0.2174,1,"{3788, 3771, 2641, 4046, 2639}"
590,1,112,0,0.1092983,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(3/2),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)}{3 d \sqrt{\sec (c+d x)}}+\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)}{3 d \sqrt{\sec (c+d x)}}+\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) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,23,0.2174,1,"{3788, 3771, 2639, 4045, 2641}"
591,1,141,0,0.1223697,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(5/2),x]","\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)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\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 \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)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\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}",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*a^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",7,6,23,0.2609,1,"{3788, 3769, 3771, 2641, 4045, 2639}"
592,1,175,0,0.137916,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(7/2),x]","\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\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)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\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 \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\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)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\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}",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) + (2*a^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*(5*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,6,23,0.2609,1,"{3788, 3769, 3771, 2639, 4045, 2641}"
593,1,234,0,0.2407963,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3,x]","\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\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) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{32 a b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}","\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\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) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{32 a b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}",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) + (2*a*(5*a^2 + 9*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(21*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (32*a*b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b^2*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)","A",9,7,23,0.3043,1,"{3842, 4047, 3768, 3771, 2641, 4046, 2639}"
594,1,189,0,0.2016908,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3,x]","\frac{6 b \left(5 a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\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{8 a b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}","\frac{6 b \left(5 a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\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{8 a b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}",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 + (6*b*(5*a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b^2*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",8,7,23,0.3043,1,"{3842, 4047, 3768, 3771, 2639, 4046, 2641}"
595,1,158,0,0.188966,"\int \frac{(a+b \sec (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\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) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}{3 d}+\frac{16 a b^2 \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) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}{3 d}+\frac{16 a b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}",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) + (16*a*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",7,6,23,0.2609,1,"{3842, 4047, 3771, 2641, 4046, 2639}"
596,1,166,0,0.1940537,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(3/2),x]","-\frac{2 b \left(a^2-3 b^2\right) \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) (a+b \sec (c+d x))}{3 d \sqrt{\sec (c+d x)}}","-\frac{2 b \left(a^2-3 b^2\right) \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) (a+b \sec (c+d x))}{3 d \sqrt{\sec (c+d x)}}",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) - (2*b*(a^2 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,23,0.2609,1,"{3841, 4047, 3771, 2641, 4046, 2639}"
597,1,156,0,0.1927982,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(5/2),x]","\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{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}","\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{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}",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 + (8*a^2*b*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",7,6,23,0.2609,1,"{3841, 4047, 3771, 2639, 4045, 2641}"
598,1,199,0,0.2279493,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(7/2),x]","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\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) (a+b \sec (c+d x))}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\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) (a+b \sec (c+d x))}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}",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) + (32*a^2*b*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a*(5*a^2 + 21*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",8,7,23,0.3043,1,"{3841, 4047, 3769, 3771, 2641, 4045, 2639}"
599,1,234,0,0.240522,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(9/2),x]","\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(15 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 \left(7 a^2+27 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a^2 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(15 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 \left(7 a^2+27 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a^2 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a*(7*a^2 + 27*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*b*(15*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (40*a^2*b*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a*(7*a^2 + 27*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*b*(15*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,7,23,0.3043,1,"{3841, 4047, 3769, 3771, 2639, 4045, 2641}"
600,1,287,0,0.4093963,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4,x]","\frac{14 b^2 \left(7 a^2+b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{8 a b \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(54 a^2 b^2+15 a^4+7 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{8 a b \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{2 \left(54 a^2 b^2+15 a^4+7 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)}{15 d}+\frac{44 a b^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}","\frac{14 b^2 \left(7 a^2+b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{8 a b \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(54 a^2 b^2+15 a^4+7 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{8 a b \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{2 \left(54 a^2 b^2+15 a^4+7 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)}{15 d}+\frac{44 a b^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}",1,"(-2*(15*a^4 + 54*a^2*b^2 + 7*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(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*(15*a^4 + 54*a^2*b^2 + 7*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a*b*(7*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (14*b^2*(7*a^2 + b^2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (44*a*b^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*b^2*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)","A",10,8,23,0.3478,1,"{3842, 4076, 4047, 3768, 3771, 2641, 4046, 2639}"
601,1,247,0,0.3659862,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4,x]","\frac{2 b^2 \left(39 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{8 a b \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(42 a^2 b^2+21 a^4+5 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)}{21 d}-\frac{8 a b \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{36 a b^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}","\frac{2 b^2 \left(39 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{8 a b \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(42 a^2 b^2+21 a^4+5 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)}{21 d}-\frac{8 a b \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{36 a b^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}",1,"(-8*a*b*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^4 + 42*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*(5*a^2 + 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*(39*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (36*a*b^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b^2*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)","A",9,8,23,0.3478,1,"{3842, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
602,1,209,0,0.3491706,"\int \frac{(a+b \sec (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^4/Sqrt[Sec[c + d*x]],x]","\frac{2 b^2 \left(29 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \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{2 \left(-30 a^2 b^2+5 a^4-3 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)}{5 d}+\frac{28 a b^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}","\frac{2 b^2 \left(29 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \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{2 \left(-30 a^2 b^2+5 a^4-3 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)}{5 d}+\frac{28 a b^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}",1,"(2*(5*a^4 - 30*a^2*b^2 - 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(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*(29*a^2 + 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (28*a*b^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b^2*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)","A",8,7,23,0.3043,1,"{3842, 4076, 4047, 3771, 2641, 4046, 2639}"
603,1,208,0,0.3486073,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(3/2),x]","-\frac{2 b^2 \left(a^2-b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 \left(18 a^2 b^2+a^4+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 a b \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) (a+b \sec (c+d x))^2}{3 d \sqrt{\sec (c+d x)}}","-\frac{2 b^2 \left(a^2-b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 \left(18 a^2 b^2+a^4+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 a b \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) (a+b \sec (c+d x))^2}{3 d \sqrt{\sec (c+d x)}}",1,"(8*a*b*(a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^4 + 18*a^2*b^2 + b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a*b*(a^2 - 6*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(a^2 - b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,23,0.3043,1,"{3841, 4076, 4047, 3771, 2641, 4046, 2639}"
604,1,207,0,0.3694296,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(5/2),x]","-\frac{2 b^2 \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \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 \left(30 a^2 b^2+3 a^4-5 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)}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}","-\frac{2 b^2 \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \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 \left(30 a^2 b^2+3 a^4-5 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)}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}",1,"(2*(3*a^4 + 30*a^2*b^2 - 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (28*a^3*b*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",8,7,23,0.3043,1,"{3841, 4074, 4047, 3771, 2641, 4046, 2639}"
605,1,211,0,0.3627489,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(7/2),x]","\frac{2 a^2 \left(5 a^2+39 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(42 a^2 b^2+5 a^4+21 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)}{21 d}+\frac{8 a b \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{36 a^3 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 \left(5 a^2+39 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(42 a^2 b^2+5 a^4+21 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)}{21 d}+\frac{8 a b \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{36 a^3 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(8*a*b*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^4 + 42*a^2*b^2 + 21*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (36*a^3*b*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a^2*(5*a^2 + 39*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",8,7,23,0.3043,1,"{3841, 4074, 4047, 3771, 2639, 4045, 2641}"
606,1,245,0,0.4025992,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(9/2),x]","\frac{14 a^2 \left(a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{8 a b \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 \left(54 a^2 b^2+7 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)}{15 d}+\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}","\frac{14 a^2 \left(a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{8 a b \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 \left(54 a^2 b^2+7 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)}{15 d}+\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*(7*a^4 + 54*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (44*a^3*b*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (14*a^2*(a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a*b*(5*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,8,23,0.3478,1,"{3841, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
607,1,289,0,0.4595028,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]","\frac{2 a^2 \left(9 a^2+59 b^2\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(7 a^2+9 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(330 a^2 b^2+45 a^4+77 b^4\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(330 a^2 b^2+45 a^4+77 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)}{231 d}+\frac{8 a b \left(7 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)}{15 d}+\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 \left(9 a^2+59 b^2\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(7 a^2+9 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(330 a^2 b^2+45 a^4+77 b^4\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(330 a^2 b^2+45 a^4+77 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)}{231 d}+\frac{8 a b \left(7 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)}{15 d}+\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(8*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (52*a^3*b*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a^2*(9*a^2 + 59*b^2)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (8*a*b*(7*a^2 + 9*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",10,8,23,0.3478,1,"{3841, 4074, 4047, 3769, 3771, 2639, 4045, 2641}"
608,1,188,0,0.5109546,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x]),x]","\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 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)}{b^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b 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 b d}","\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 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)}{b^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b 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 b d}",1,"(2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b*d) + (2*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) - (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)","A",10,9,23,0.3913,1,"{3851, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
609,1,117,0,0.1638445,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x]),x]","-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b 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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b 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]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",6,6,23,0.2609,1,"{3850, 3768, 3771, 2639, 3849, 2805}"
610,1,49,0,0.0915231,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d)","A",2,2,23,0.08696,1,"{3849, 2805}"
611,1,93,0,0.1501506,"\int \frac{\sqrt{\sec (c+d x)}}{a+b \sec (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)","A",4,4,23,0.1739,1,"{3848, 2803, 2641, 2805}"
612,1,135,0,0.2073206,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{2 b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \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^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \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]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)","A",8,7,23,0.3043,1,"{3852, 3849, 2805, 3787, 3771, 2639, 2641}"
613,1,172,0,0.3664784,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),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 a^3 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 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)}{3 a d \sqrt{\sec (c+d 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 a^3 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 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)}{3 a d \sqrt{\sec (c+d x)}}",1,"(-2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",9,8,23,0.3478,1,"{3853, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
614,1,342,0,0.9441331,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(5 a^2-2 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left(a^2-b^2\right)}+\frac{\left(5 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)}{3 b^2 d \left(a^2-b^2\right)}+\frac{a \left(5 a^2-4 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^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(5 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(5 a^2-2 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left(a^2-b^2\right)}+\frac{\left(5 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)}{3 b^2 d \left(a^2-b^2\right)}+\frac{a \left(5 a^2-4 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^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(5 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"(a*(5*a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) + (a^2*(5*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - (a*(5*a^2 - 4*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",11,9,23,0.3913,1,"{3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
615,1,279,0,0.6432406,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\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{\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 \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\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{\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 \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"-(((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*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
616,1,214,0,0.4017691,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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{\left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b 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+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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{\left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b 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) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b*(a + b)^2*d) - (a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3845, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
617,1,208,0,0.3575567,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2,x]","\frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\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{\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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a 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+b \sec (c+d x))}-\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{\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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a 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)) - (b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*(a + b)^2*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3844, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
618,1,227,0,0.3691991,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{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{b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}","-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{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{b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"(b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3843, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
619,1,244,0,0.439661,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[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+b \sec (c+d x))}-\frac{b \left(4 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)}{a^3 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^2 \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}","\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b \left(4 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)}{a^3 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^2 \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((2*a^2 - 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*(4*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,23,0.3478,1,"{3847, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
620,1,304,0,0.6803837,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left(2 a^2-5 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\left(16 a^2 b^2+2 a^4-15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{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^3 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}","\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left(2 a^2-5 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\left(16 a^2 b^2+2 a^4-15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{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^3 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}",1,"-((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^4 + 16*a^2*b^2 - 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3847, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
621,1,388,0,0.961193,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^3,x]","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(5 a^2-11 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(-29 a^2 b^2+15 a^4+8 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 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-38 a^2 b^2+15 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(5 a^2-11 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(-29 a^2 b^2+15 a^4+8 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 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(-38 a^2 b^2+15 a^4+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"-((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - (a*(5*a^2 - 11*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*(15*a^4 - 38*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) - (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(5*a^2 - 11*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
622,1,315,0,0.7035747,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^3,x]","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(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 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 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(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 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 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 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) + ((a^2 - 7*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 - 2*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (3*a^2*(a^2 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
623,1,313,0,0.6827913,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^3,x]","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{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+b \sec (c+d x))}+\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 a 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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{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+b \sec (c+d x))}+\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 a 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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b 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) + (3*(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) + ((a^4 - 10*a^2*b^2 - 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*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*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3845, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
624,1,306,0,0.6456426,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3,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+b \sec (c+d x))}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{b \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^2 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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^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+b \sec (c+d x))}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{b \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^2 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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 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) - (b*(7*a^2 - 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) + ((3*a^4 + 10*a^2*b^2 - b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*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*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3844, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
625,1,323,0,0.667242,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^3} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^3,x]","-\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+b \sec (c+d x))}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(-5 a^2 b^2+8 a^4+3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{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 b \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}","-\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+b \sec (c+d x))}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(-5 a^2 b^2+8 a^4+3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{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 b \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 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) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*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*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3843, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
626,1,342,0,0.7543744,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{b^2 \left(11 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 b \left(-11 a^2 b^2+8 a^4+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 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^2 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}","\frac{b^2 \left(11 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 b \left(-11 a^2 b^2+8 a^4+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 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^2 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((8*a^4 - 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) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{3847, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
627,1,406,0,1.0205959,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\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 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(128 a^4 b^2-223 a^2 b^4+8 a^6+105 b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{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^3 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}","\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(128 a^4 b^2-223 a^2 b^4+8 a^6+105 b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{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^3 \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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 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^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[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*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{3847, 4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
628,1,237,0,0.6529061,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,25,0.4800,1,"{3855, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
629,1,138,0,0.3541347,"\int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Int[Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3854, 3858, 2663, 2661, 3859, 2807, 2805}"
630,1,67,0,0.0971235,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3856, 2655, 2653}"
631,1,192,0,0.3773916,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3857, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
632,1,244,0,0.6575037,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/2),x]","-\frac{4 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a d \sqrt{\sec (c+d x)}}","-\frac{4 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a d \sqrt{\sec (c+d x)}}",1,"(-4*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 - 2*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3857, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
633,1,305,0,0.8468792,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]","\frac{2 \left(25 a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(-17 a^2 b^2+25 a^4-8 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(19 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(25 a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(-17 a^2 b^2+25 a^4-8 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(19 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(25*a^4 - 17*a^2*b^2 - 8*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(19*a^2 + 8*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2 - 4*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])","A",10,9,25,0.3600,1,"{3857, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
634,1,299,0,0.994374,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{5 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{7 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{5 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{7 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(7*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - (5*a*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (5*a*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",13,13,25,0.5200,1,"{3866, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
635,1,249,0,0.726712,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\left(2 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{\left(2 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((2*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (3*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,25,0.4800,1,"{3866, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
636,1,209,0,0.498619,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{2 b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",11,11,25,0.4400,1,"{3868, 3856, 2655, 2653, 3854, 3858, 2663, 2661, 3859, 2807, 2805}"
637,1,187,0,0.4059118,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (8*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3864, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
638,1,240,0,0.6148308,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2),x]","\frac{2 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sqrt{\sec (c+d x)}}","\frac{2 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sqrt{\sec (c+d x)}}",1,"(2*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(5*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 + b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3864, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
639,1,303,0,0.8713186,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 \left(-31 a^2 b^2+25 a^4+6 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{16 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 \left(-31 a^2 b^2+25 a^4+6 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{16 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(41*a^2 - 3*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (16*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]])","A",10,9,25,0.3600,1,"{3864, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
640,1,369,0,1.3475698,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}",1,"(b*(59*a^2 + 16*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + (5*a*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((33*a^2 + 16*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((33*a^2 + 16*b^2)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (13*a*b*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (b^2*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",14,13,25,0.5200,1,"{3842, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
641,1,314,0,1.0709405,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}-\frac{9 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}-\frac{9 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(a*(8*a^2 + 11*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - (9*a*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (9*a*b*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b^2*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",13,13,25,0.5200,1,"{3842, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
642,1,263,0,0.7802797,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{b \left(4 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{5 a b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{b \left(4 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{5 a b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(b*(4*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (5*a*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a^2 - b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,25,0.4800,1,"{3842, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
643,1,262,0,0.780261,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^3 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^3 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*a*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (14*a*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",12,12,25,0.4800,1,"{3841, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
644,1,239,0,0.6898518,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2),x]","\frac{16 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{22 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}","\frac{16 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{22 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}",1,"(16*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 23*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (22*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
645,1,303,0,0.9356253,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2),x]","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(-2 a^2 b^2+5 a^4-3 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{6 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(-2 a^2 b^2+5 a^4-3 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{6 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(21*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(29*a^2 + 3*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(21*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",10,9,25,0.3600,1,"{3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
646,1,363,0,1.2539238,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d \sqrt{\sec (c+d x)}}+\frac{4 b \left(-62 a^2 b^2+57 a^4+5 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 b^2+147 a^4-10 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{38 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d \sqrt{\sec (c+d x)}}+\frac{4 b \left(-62 a^2 b^2+57 a^4+5 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 b^2+147 a^4-10 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{38 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (38*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]])","A",11,9,25,0.3600,1,"{3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
647,1,312,0,0.8880678,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(7/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b^2 d}+\frac{3 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b^2 d}+\frac{3 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}",1,"-(a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (3*a*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",13,13,25,0.5200,1,"{3860, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
648,1,246,0,0.6172798,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b d}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b d}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",12,12,25,0.4800,1,"{3860, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
649,1,68,0,0.1815619,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(3/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3859, 2807, 2805}"
650,1,67,0,0.0953826,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3858, 2663, 2661}"
651,1,142,0,0.2424804,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}",1,"(-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3862, 3856, 2655, 2653, 3858, 2663, 2661}"
652,1,195,0,0.3618546,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d \sqrt{\sec (c+d x)}}","\frac{2 \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d \sqrt{\sec (c+d x)}}",1,"(2*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (4*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3863, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
653,1,249,0,0.5671004,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{8 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{8 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*b*(7*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 8*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (8*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3863, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
654,1,345,0,0.9974182,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{\left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{\left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (3*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((3*a^2 - b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",13,13,25,0.5200,1,"{3845, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
655,1,206,0,0.5201697,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(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 \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3845, 4108, 3859, 2807, 2805, 21, 3856, 2655, 2653}"
656,1,126,0,0.1641824,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/((a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3844, 21, 3856, 2655, 2653}"
657,1,200,0,0.3795138,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^(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 \sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}","-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3843, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
658,1,214,0,0.4340043,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[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 \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{a+b \sec (c+d x)}}",1,"(-4*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 2*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{3847, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
659,1,289,0,0.6719517,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(5*a^2 - 8*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3847, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
660,1,360,0,0.9663532,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-8*b*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(5*a^4*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 6*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (2*b*(3*a^2 - 8*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",10,9,25,0.3600,1,"{3847, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
661,1,458,0,1.4132072,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^2 \left(5 a^2-9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\left(-26 a^2 b^2+15 a^4+3 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2-3 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-26 a^2 b^2+15 a^4+3 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{5 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^2 \left(5 a^2-9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\left(-26 a^2 b^2+15 a^4+3 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2-3 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-26 a^2 b^2+15 a^4+3 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{5 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{a+b \sec (c+d x)}}",1,"((5*a^2 - 3*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (5*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((15*a^4 - 26*a^2*b^2 + 3*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(5*a^2 - 9*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + ((15*a^4 - 26*a^2*b^2 + 3*b^4)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)","A",14,14,25,0.5600,1,"{3845, 4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
662,1,370,0,1.103095,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^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 \sec (c+d x)}}-\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^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 \sec (c+d x)}}-\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}",1,"(-2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*(3*a^2 - 7*b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^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*Sec[c + d*x]])","A",13,13,25,0.5200,1,"{3845, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
663,1,277,0,0.6649366,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","-\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (8*b*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3845, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
664,1,281,0,0.6139082,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{4 \left(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 \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{4 \left(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 \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2 + b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*(a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3844, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
665,1,302,0,0.6447606,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 b \left(5 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 \sec (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","-\frac{2 b \left(5 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 \sec (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(3*a^2 - 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(3*a^2 - b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3843, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
666,1,317,0,0.7341477,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[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 \sec (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 \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{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 \sec (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 \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*b*(9*a^2 - 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[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*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{3847, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
667,1,391,0,1.0220033,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[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 \sqrt{\sec (c+d x)} \sqrt{a+b \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 \sec (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 \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{2 \left(16 a^2 b^2+a^4-16 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{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 \sqrt{\sec (c+d x)} \sqrt{a+b \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 \sec (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 \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{2 \left(16 a^2 b^2+a^4-16 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^4 + 16*a^2*b^2 - 16*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[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*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",10,10,25,0.4000,1,"{3847, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
668,1,474,0,1.3533135,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{8 b^2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-71 a^2 b^2+3 a^4+48 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 b \left(-49 a^2 b^2+7 a^4+32 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{2 b \left(116 a^2 b^2+17 a^4-128 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(55 a^4 b^2-212 a^2 b^4+9 a^6+128 b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{8 b^2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-71 a^2 b^2+3 a^4+48 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 b \left(-49 a^2 b^2+7 a^4+32 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{2 b \left(116 a^2 b^2+17 a^4-128 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(55 a^4 b^2-212 a^2 b^4+9 a^6+128 b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*b*(17*a^4 + 116*a^2*b^2 - 128*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (8*b^2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 - 71*a^2*b^2 + 48*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",11,10,25,0.4000,1,"{3847, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
669,1,122,0,0.1799074,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{2+3 \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[2 + 3*Sec[c + d*x]]),x]","\frac{\sqrt{5} \sqrt{3 \sec (c+d x)+2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}","\frac{\sqrt{5} \sqrt{3 \sec (c+d x)+2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}",1,"(-3*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]]) + (Sqrt[5]*EllipticE[(c + d*x)/2, 4/5]*Sqrt[2 + 3*Sec[c + d*x]])/(d*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2653, 3858, 2661}"
670,1,109,0,0.1756713,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-2+3 \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[-2 + 3*Sec[c + d*x]]),x]","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}-\frac{\sqrt{3 \sec (c+d x)-2} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}-\frac{\sqrt{3 \sec (c+d x)-2} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"(3*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]]) - (EllipticE[(c + d*x)/2, -4]*Sqrt[-2 + 3*Sec[c + d*x]])/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2653, 3858, 2661}"
671,1,108,0,0.198349,"\int \frac{1}{\sqrt{2-3 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[2 - 3*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}+\frac{\sqrt{2-3 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}+\frac{\sqrt{2-3 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"(EllipticE[(c + d*x)/2, -4]*Sqrt[2 - 3*Sec[c + d*x]])/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3862, 3856, 2655, 2653, 3858, 2663, 2661}"
672,1,123,0,0.2042023,"\int \frac{1}{\sqrt{-2-3 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[-2 - 3*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}-\frac{\sqrt{5} \sqrt{-3 \sec (c+d x)-2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}","-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}-\frac{\sqrt{5} \sqrt{-3 \sec (c+d x)-2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}",1,"-((Sqrt[5]*EllipticE[(c + d*x)/2, 4/5]*Sqrt[-2 - 3*Sec[c + d*x]])/(d*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) - (3*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])","A",7,7,25,0.2800,1,"{3862, 3856, 2655, 2653, 3858, 2663, 2661}"
673,1,127,0,0.1712972,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{3+2 \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[3 + 2*Sec[c + d*x]]),x]","\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)+3} E\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 d \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)}}-\frac{4 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)+3}}","\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)+3} E\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 d \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)}}-\frac{4 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)+3}}",1,"(-4*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]]) + (2*Sqrt[5]*EllipticE[(c + d*x)/2, 6/5]*Sqrt[3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2653, 3858, 2661}"
674,1,113,0,0.1766486,"\int \frac{1}{\sqrt{3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{4 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3-2 \sec (c+d x)}}+\frac{2 \sqrt{3-2 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}","\frac{4 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3-2 \sec (c+d x)}}+\frac{2 \sqrt{3-2 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}",1,"(2*EllipticE[(c + d*x)/2, 6]*Sqrt[3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[3 - 2*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2653, 3858, 2661}"
675,1,129,0,0.1741125,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-3+2 \sec (c+d x)}} \, dx","Int[1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 + 2*Sec[c + d*x]]),x]","\frac{4 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)-3}}-\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)-3} E\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}","\frac{4 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)-3}}-\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)-3} E\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"(4*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[-3 + 2*Sec[c + d*x]]) - (2*Sqrt[5]*EllipticE[(c + Pi + d*x)/2, 6/5]*Sqrt[-3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2654, 3858, 2662}"
676,1,115,0,0.1745626,"\int \frac{1}{\sqrt{-3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Int[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{4 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-2 \sec (c+d x)-3}}-\frac{2 \sqrt{-2 \sec (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}","-\frac{4 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-2 \sec (c+d x)-3}}-\frac{2 \sqrt{-2 \sec (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}",1,"(-2*EllipticE[(c + Pi + d*x)/2, 6]*Sqrt[-3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 - 2*Sec[c + d*x]])","A",5,5,25,0.2000,1,"{3862, 3856, 2654, 3858, 2662}"
677,1,61,0,0.0568292,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2+3 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[2 + 3*Sec[c + d*x]],x]","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}",1,"(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2661}"
678,1,54,0,0.0566,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2+3 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[-2 + 3*Sec[c + d*x]],x]","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}",1,"(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2661}"
679,1,54,0,0.0692775,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2-3 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[2 - 3*Sec[c + d*x]],x]","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}",1,"(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3858, 2663, 2661}"
680,1,61,0,0.0700731,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2-3 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[-2 - 3*Sec[c + d*x]],x]","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}",1,"(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])","A",3,3,25,0.1200,1,"{3858, 2663, 2661}"
681,1,61,0,0.0570815,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3+2 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[3 + 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}}","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}}",1,"(2*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2661}"
682,1,54,0,0.0570182,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3-2 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[3 - 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{d \sqrt{3-2 \sec (c+d x)}}","\frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{d \sqrt{3-2 \sec (c+d x)}}",1,"(2*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[3 - 2*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2661}"
683,1,62,0,0.0571196,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3+2 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[-3 + 2*Sec[c + d*x]],x]","\frac{2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)-3}}","\frac{2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)-3}}",1,"(2*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-3 + 2*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2662}"
684,1,55,0,0.0580804,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3-2 \sec (c+d x)}} \, dx","Int[Sqrt[Sec[c + d*x]]/Sqrt[-3 - 2*Sec[c + d*x]],x]","\frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{d \sqrt{-2 \sec (c+d x)-3}}","\frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{d \sqrt{-2 \sec (c+d x)-3}}",1,"(2*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-3 - 2*Sec[c + d*x]])","A",2,2,25,0.08000,1,"{3858, 2662}"
685,1,105,0,0.0965397,"\int \sec (c+d x) \sqrt[3]{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(1/3),x]","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
686,0,0,0,0.0108199,"\int \sqrt[3]{a+b \sec (c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(1/3),x]","\int \sqrt[3]{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sqrt[3]{a+b \sec (c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(1/3), x]","A",0,0,0,0,-1,"{}"
687,1,362,0,0.6682297,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(2/3),x]","\frac{a \left(18 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\left(23 a^2 b^2+9 a^4-32 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{55 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \left(9 a^2+32 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{220 b^2 d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{44 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}","\frac{a \left(18 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\left(23 a^2 b^2+9 a^4-32 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{55 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \left(9 a^2+32 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{220 b^2 d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{44 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}",1,"(3*(9*a^2 + 32*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(220*b^2*d) - (9*a*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(44*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(11*b*d) + (a*(18*a^2 + 49*b^2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(110*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - ((9*a^4 + 23*a^2*b^2 - 32*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(55*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",10,7,23,0.3043,1,"{3865, 4082, 4002, 4007, 3834, 139, 138}"
688,1,305,0,0.4702801,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(2/3),x]","-\frac{\left(6 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{40 b d}","-\frac{\left(6 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{40 b d}",1,"(-9*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*b*d) + (3*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b*d) - ((6*a^2 - 25*b^2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (3*a*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(10*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",9,6,23,0.2609,1,"{3840, 4002, 4007, 3834, 139, 138}"
689,1,260,0,0.3382582,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(2/3),x]","-\frac{2 \sqrt{2} \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{2 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}","-\frac{2 \sqrt{2} \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{2 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}",1,"(3*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (2*Sqrt[2]*a*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (2*Sqrt[2]*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(5*b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",8,5,23,0.2174,1,"{3835, 4007, 3834, 139, 138}"
690,1,105,0,0.0829725,"\int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(2/3),x]","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
691,0,0,0,0.0117092,"\int (a+b \sec (c+d x))^{2/3} \, dx","Int[(a + b*Sec[c + d*x])^(2/3),x]","\int (a+b \sec (c+d x))^{2/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{2/3},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(2/3), x]","A",0,0,0,0,-1,"{}"
692,1,108,0,0.0883491,"\int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(4/3),x]","\frac{\sqrt{2} (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}","\frac{\sqrt{2} (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"(Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
693,0,0,0,0.0114037,"\int (a+b \sec (c+d x))^{4/3} \, dx","Int[(a + b*Sec[c + d*x])^(4/3),x]","\int (a+b \sec (c+d x))^{4/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{4/3},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(4/3), x]","A",0,0,0,0,-1,"{}"
694,1,412,0,0.8252406,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/3),x]","\frac{\left(164 a^2 b^2+36 a^4+605 b^4\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{616 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(79 a^2 b^2+18 a^4-97 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{308 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \left(18 a^2+121 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{1232 b^2 d}+\frac{3 a \left(18 a^2+97 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{1232 b^2 d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{77 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{8/3}}{14 b d}","\frac{\left(164 a^2 b^2+36 a^4+605 b^4\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{616 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(79 a^2 b^2+18 a^4-97 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{308 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \left(18 a^2+121 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{1232 b^2 d}+\frac{3 a \left(18 a^2+97 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{1232 b^2 d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{77 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{8/3}}{14 b d}",1,"(3*a*(18*a^2 + 97*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(1232*b^2*d) + (3*(18*a^2 + 121*b^2)*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(1232*b^2*d) - (9*a*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(77*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(14*b*d) + ((36*a^4 + 164*a^2*b^2 + 605*b^4)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(616*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(18*a^4 + 79*a^2*b^2 - 97*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(308*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",11,7,23,0.3043,1,"{3865, 4082, 4002, 4007, 3834, 139, 138}"
695,1,356,0,0.6528388,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/3),x]","-\frac{a \left(30 a^2-373 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{220 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{\left(-79 a^2 b^2+15 a^4+64 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{3 \left(15 a^2-64 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{440 b d}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{11 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{88 b d}","-\frac{a \left(30 a^2-373 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{220 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{\left(-79 a^2 b^2+15 a^4+64 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{3 \left(15 a^2-64 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{440 b d}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{11 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{88 b d}",1,"(-3*(15*a^2 - 64*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(440*b*d) - (9*a*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(88*b*d) + (3*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*b*d) - (a*(30*a^2 - 373*b^2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(220*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + ((15*a^4 - 79*a^2*b^2 + 64*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(110*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",10,6,23,0.2609,1,"{3840, 4002, 4007, 3834, 139, 138}"
696,1,299,0,0.4566052,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/3),x]","\frac{\left(2 a^2+5 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{4 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{8 d}","\frac{\left(2 a^2+5 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{4 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{8 d}",1,"(3*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*d) + (3*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*d) + ((2*a^2 + 5*b^2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(4*Sqrt[2]*b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(2*Sqrt[2]*b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",9,6,23,0.2609,1,"{3835, 4002, 4007, 3834, 139, 138}"
697,1,108,0,0.0887916,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3),x]","\frac{\sqrt{2} (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}","\frac{\sqrt{2} (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"(Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -5/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
698,0,0,0,0.0113138,"\int (a+b \sec (c+d x))^{5/3} \, dx","Int[(a + b*Sec[c + d*x])^(5/3),x]","\int (a+b \sec (c+d x))^{5/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{5/3},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/3), x]","A",0,0,0,0,-1,"{}"
699,1,313,0,0.4882157,"\int \frac{\sec ^4(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(1/3),x]","-\frac{a \left(9 a^2+11 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\left(18 a^2+25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d}","-\frac{a \left(9 a^2+11 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\left(18 a^2+25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d}",1,"(-9*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*b*d) + ((18*a^2 + 25*b^2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(9*a^2 + 11*b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(10*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",9,6,23,0.2609,1,"{3865, 4082, 4007, 3834, 139, 138}"
700,1,265,0,0.3322335,"\int \frac{\sec ^3(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(1/3),x]","\frac{\sqrt{2} \left(3 a^2+2 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{3 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 b d}","\frac{\sqrt{2} \left(3 a^2+2 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{3 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 b d}",1,"(3*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d) - (3*Sqrt[2]*a*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(3*a^2 + 2*b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(5*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",8,5,23,0.2174,1,"{3840, 4007, 3834, 139, 138}"
701,1,219,0,0.2286417,"\int \frac{\sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(1/3),x]","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\sqrt{2} a \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\sqrt{2} a \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",7,4,23,0.1739,1,"{3838, 3834, 139, 138}"
702,1,105,0,0.0757841,"\int \frac{\sec (c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(1/3),x]","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
703,0,0,0,0.0107156,"\int \frac{1}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(-1/3),x]","\int \frac{1}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(-1/3), x]","A",0,0,0,0,-1,"{}"
704,1,105,0,0.0837088,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(2/3),x]","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
705,0,0,0,0.0108436,"\int \frac{1}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[(a + b*Sec[c + d*x])^(-2/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{2/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{2/3}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(-2/3), x]","A",0,0,0,0,-1,"{}"
706,1,110,0,0.0876113,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{4/3}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(4/3),x]","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, 4/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/((a + b)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
707,0,0,0,0.0111019,"\int \frac{1}{(a+b \sec (c+d x))^{4/3}} \, dx","Int[(a + b*Sec[c + d*x])^(-4/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{4/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{4/3}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(-4/3), x]","A",0,0,0,0,-1,"{}"
708,1,378,0,0.5604084,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/3),x]","-\frac{a \left(9 a^2-7 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{\left(-10 a^2 b^2+9 a^4-b^4\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}+\frac{3 \left(3 a^2-b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)}","-\frac{a \left(9 a^2-7 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{\left(-10 a^2 b^2+9 a^4-b^4\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}+\frac{3 \left(3 a^2-b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)}",1,"(-3*a^2*Sec[c + d*x]*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3)) + (3*(3*a^2 - b^2)*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b^2*(a^2 - b^2)*d) - (a*(9*a^2 - 7*b^2)*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(2*Sqrt[2]*b^3*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + ((9*a^4 - 10*a^2*b^2 - b^4)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(2*Sqrt[2]*b^3*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",9,6,23,0.2609,1,"{3845, 4082, 4007, 3834, 139, 138}"
709,1,307,0,0.3806144,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/3),x]","-\frac{a \left(3 a^2-4 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\left(3 a^2-2 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}-\frac{3 a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}","-\frac{a \left(3 a^2-4 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\left(3 a^2-2 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}-\frac{3 a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}",1,"(-3*a^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3)) + ((3*a^2 - 2*b^2)*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(Sqrt[2]*b^2*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (a*(3*a^2 - 4*b^2)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(Sqrt[2]*b^2*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",8,5,23,0.2174,1,"{3839, 4007, 3834, 139, 138}"
710,1,289,0,0.3593299,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/3),x]","\frac{\left(a^2-2 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}-\frac{a \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{3 a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}","\frac{\left(a^2-2 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}-\frac{a \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{3 a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}",1,"(3*a*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3)) - (a*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(Sqrt[2]*b*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + ((a^2 - 2*b^2)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(Sqrt[2]*b*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",8,5,23,0.2174,1,"{3836, 4007, 3834, 139, 138}"
711,1,110,0,0.0865168,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Int[Sec[c + d*x]/(a + b*Sec[c + d*x])^(5/3),x]","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, 5/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/((a + b)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",3,3,21,0.1429,1,"{3834, 139, 138}"
712,0,0,0,0.0114356,"\int \frac{1}{(a+b \sec (c+d x))^{5/3}} \, dx","Int[(a + b*Sec[c + d*x])^(-5/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{5/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{5/3}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(-5/3), x]","A",0,0,0,0,-1,"{}"
713,1,174,0,0.2570512,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x]),x]","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sqrt[3]{\sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\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{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sqrt[3]{\sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\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{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"(a*AppellF1[1/2, -1/6, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)) - (b*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d)","A",6,4,23,0.1739,1,"{3869, 2823, 3189, 429}"
714,1,174,0,0.2623466,"\int \frac{\sqrt[3]{\sec (c+d x)}}{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x]),x]","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sec ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sec ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"(a*AppellF1[1/2, -1/3, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)) - (b*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d)","A",6,4,23,0.1739,1,"{3869, 2823, 3189, 429}"
715,1,174,0,0.2404858,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Int[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])),x]","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sqrt[3]{\sec (c+d x)}}","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sqrt[3]{\sec (c+d x)}}",1,"-((b*AppellF1[1/2, -1/6, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3))) + (a*AppellF1[1/2, -2/3, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d)","A",6,4,23,0.1739,1,"{3869, 2823, 3189, 429}"
716,1,174,0,0.2401657,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])),x]","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};-\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sec ^{\frac{2}{3}}(c+d x)}","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};-\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^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)} \sec ^{\frac{2}{3}}(c+d x)}",1,"-((b*AppellF1[1/2, -1/3, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3))) + (a*AppellF1[1/2, -5/6, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d)","A",6,4,23,0.1739,1,"{3869, 2823, 3189, 429}"
717,0,0,0,0.0521502,"\int \sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
718,0,0,0,0.0521657,"\int \sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
719,0,0,0,0.0516693,"\int \sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
720,0,0,0,0.0522901,"\int \sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
721,0,0,0,0.0525696,"\int \sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Int[Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
722,0,0,0,0.052302,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3), x]","A",0,0,0,0,-1,"{}"
723,0,0,0,0.0520735,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3), x]","A",0,0,0,0,-1,"{}"
724,0,0,0,0.0505842,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3), x]","A",0,0,0,0,-1,"{}"
725,0,0,0,0.0505002,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3), x]","A",0,0,0,0,-1,"{}"
726,0,0,0,0.0506645,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x]","A",0,0,0,0,-1,"{}"
727,0,0,0,0.0591549,"\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
728,0,0,0,0.0598057,"\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
729,0,0,0,0.05696,"\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
730,0,0,0,0.0567228,"\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
731,0,0,0,0.0573267,"\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
732,0,0,0,0.0579205,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3), x]","A",0,0,0,0,-1,"{}"
733,0,0,0,0.0572684,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3), x]","A",0,0,0,0,-1,"{}"
734,0,0,0,0.0574033,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3), x]","A",0,0,0,0,-1,"{}"
735,0,0,0,0.0565043,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3), x]","A",0,0,0,0,-1,"{}"
736,0,0,0,0.05757,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x]","A",0,0,0,0,-1,"{}"
737,0,0,0,0.0560519,"\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
738,0,0,0,0.0563213,"\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
739,0,0,0,0.0565535,"\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
740,0,0,0,0.0561414,"\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
741,0,0,0,0.0553584,"\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
742,0,0,0,0.0576654,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3), x]","A",0,0,0,0,-1,"{}"
743,0,0,0,0.0574173,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3), x]","A",0,0,0,0,-1,"{}"
744,0,0,0,0.0561252,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x]","A",0,0,0,0,-1,"{}"
745,0,0,0,0.0558461,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3), x]","A",0,0,0,0,-1,"{}"
746,0,0,0,0.0576089,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3), x]","A",0,0,0,0,-1,"{}"
747,0,0,0,0.0520441,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
748,0,0,0,0.0522877,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
749,0,0,0,0.0518766,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
750,0,0,0,0.0523929,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
751,0,0,0,0.0520848,"\int \frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
752,0,0,0,0.0531695,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
753,0,0,0,0.0522012,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
754,0,0,0,0.0528158,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
755,0,0,0,0.051882,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
756,0,0,0,0.0535342,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",0,0,0,0,-1,"{}"
757,0,0,0,0.0592837,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
758,0,0,0,0.0588555,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
759,0,0,0,0.0586029,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
760,0,0,0,0.0582848,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
761,0,0,0,0.0591886,"\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
762,0,0,0,0.0579604,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",0,0,0,0,-1,"{}"
763,0,0,0,0.0597639,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",0,0,0,0,-1,"{}"
764,0,0,0,0.0596298,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",0,0,0,0,-1,"{}"
765,0,0,0,0.0592022,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",0,0,0,0,-1,"{}"
766,0,0,0,0.0594766,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",0,0,0,0,-1,"{}"
767,0,0,0,0.0590322,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
768,0,0,0,0.0587356,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
769,0,0,0,0.059279,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
770,0,0,0,0.059908,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
771,0,0,0,0.058759,"\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
772,0,0,0,0.0600129,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",0,0,0,0,-1,"{}"
773,0,0,0,0.0593236,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",0,0,0,0,-1,"{}"
774,0,0,0,0.0598054,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",0,0,0,0,-1,"{}"
775,0,0,0,0.0592679,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",0,0,0,0,-1,"{}"
776,0,0,0,0.0596412,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Defer[Int][1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",0,0,0,0,-1,"{}"
777,1,251,0,0.3490223,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^3 \, dx","Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^3,x]","-\frac{a d \left(a^2 (n+1)+3 b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n+2)+b^2 (n+1)\right) \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n+5) \tan (e+f x) (d \sec (e+f x))^n}{f (n+1) (n+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \sec (e+f x))^n}{f (n+2)}","-\frac{a d \left(a^2 (n+1)+3 b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n+2)+b^2 (n+1)\right) \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n+5) \tan (e+f x) (d \sec (e+f x))^n}{f (n+1) (n+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \sec (e+f x))^n}{f (n+2)}",1,"-((a*d*(3*b^2*n + a^2*(1 + n))*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2])) + (b*(b^2*(1 + n) + 3*a^2*(2 + n))*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 + 2*n)*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + n)*(2 + n)) + (b^2*(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n))","A",7,5,23,0.2174,1,"{3842, 4047, 3772, 2643, 4046}"
778,1,181,0,0.1499596,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^2 \, dx","Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^2,x]","-\frac{d \left(a^2 (n+1)+b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \sec (e+f x))^n}{f (n+1)}","-\frac{d \left(a^2 (n+1)+b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \sec (e+f x))^n}{f (n+1)}",1,"-((d*(b^2*n + a^2*(1 + n))*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2])) + (2*a*b*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2]) + (b^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + n))","A",6,4,23,0.1739,1,"{3788, 3772, 2643, 4046}"
779,1,137,0,0.094441,"\int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx","Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x]),x]","\frac{b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a d \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}","\frac{b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a d \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"-((a*d*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (b*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])","A",5,3,21,0.1429,1,"{3787, 3772, 2643}"
780,1,192,0,0.3021764,"\int \frac{(d \sec (e+f x))^n}{a+b \sec (e+f x)} \, dx","Int[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"(a*AppellF1[1/2, (-1 + n)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n)/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, n/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f)","A",6,4,23,0.1739,1,"{3869, 2823, 3189, 429}"
781,1,299,0,0.4436254,"\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^2} \, dx","Int[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^2,x]","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-3}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-2}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-3}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-2}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"(a^2*AppellF1[1/2, (-3 + n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n)/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (-1 + n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((-1 + n)/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f) - (2*a*b*AppellF1[1/2, (-2 + n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f)","A",9,4,23,0.1739,1,"{3869, 2824, 3189, 429}"
782,0,0,0,0.0686343,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx","Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]","\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx","\text{Int}\left((a+b \sec (e+f x))^{3/2} (d \sec (e+f x))^n,x\right)",0,"Defer[Int][(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
783,0,0,0,0.0615866,"\int (d \sec (e+f x))^n \sqrt{a+b \sec (e+f x)} \, dx","Int[(d*Sec[e + f*x])^n*Sqrt[a + b*Sec[e + f*x]],x]","\int (d \sec (e+f x))^n \sqrt{a+b \sec (e+f x)} \, dx","\text{Int}\left(\sqrt{a+b \sec (e+f x)} (d \sec (e+f x))^n,x\right)",0,"Defer[Int][(d*Sec[e + f*x])^n*Sqrt[a + b*Sec[e + f*x]], x]","A",0,0,0,0,-1,"{}"
784,0,0,0,0.0701936,"\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[(d*Sec[e + f*x])^n/Sqrt[a + b*Sec[e + f*x]],x]","\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx","\text{Int}\left(\frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}},x\right)",0,"Defer[Int][(d*Sec[e + f*x])^n/Sqrt[a + b*Sec[e + f*x]], x]","A",0,0,0,0,-1,"{}"
785,0,0,0,0.0712641,"\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx","Int[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2),x]","\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}},x\right)",0,"Defer[Int][(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
786,0,0,0,0.041781,"\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m,x]","\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\sec ^n(e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Defer[Int][Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
787,0,0,0,0.0465105,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx","Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m,x]","\int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx","\text{Int}\left((d \sec (e+f x))^n (a+b \sec (e+f x))^m,x\right)",0,"Defer[Int][(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
788,1,273,0,0.348737,"\int \sec ^3(e+f x) (a+b \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^3*(a + b*Sec[e + f*x])^m,x]","\frac{\sqrt{2} \left(a^2+b^2 (m+1)\right) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}+\frac{\tan (e+f x) (a+b \sec (e+f x))^{m+1}}{b f (m+2)}","\frac{\sqrt{2} \left(a^2+b^2 (m+1)\right) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}+\frac{\tan (e+f x) (a+b \sec (e+f x))^{m+1}}{b f (m+2)}",1,"((a + b*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(b*f*(2 + m)) - (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Sec[e + f*x])/2, (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(b^2*f*(2 + m)*Sqrt[1 + Sec[e + f*x]]*((a + b*Sec[e + f*x])/(a + b))^m) + (Sqrt[2]*(a^2 + b^2*(1 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sec[e + f*x])/2, (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(b^2*f*(2 + m)*Sqrt[1 + Sec[e + f*x]]*((a + b*Sec[e + f*x])/(a + b))^m)","A",8,5,21,0.2381,1,"{3840, 4007, 3834, 139, 138}"
789,1,220,0,0.2212045,"\int \sec ^2(e+f x) (a+b \sec (e+f x))^m \, dx","Int[Sec[e + f*x]^2*(a + b*Sec[e + f*x])^m,x]","\frac{\sqrt{2} (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}","\frac{\sqrt{2} (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}",1,"(Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Sec[e + f*x])/2, (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(b*f*Sqrt[1 + Sec[e + f*x]]*((a + b*Sec[e + f*x])/(a + b))^m) - (Sqrt[2]*a*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sec[e + f*x])/2, (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(b*f*Sqrt[1 + Sec[e + f*x]]*((a + b*Sec[e + f*x])/(a + b))^m)","A",7,4,21,0.1905,1,"{3838, 3834, 139, 138}"
790,1,103,0,0.073033,"\int \sec (e+f x) (a+b \sec (e+f x))^m \, dx","Int[Sec[e + f*x]*(a + b*Sec[e + f*x])^m,x]","\frac{\sqrt{2} \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{f \sqrt{\sec (e+f x)+1}}","\frac{\sqrt{2} \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sec[e + f*x])/2, (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]]*((a + b*Sec[e + f*x])/(a + b))^m)","A",3,3,19,0.1579,1,"{3834, 139, 138}"
791,0,0,0,0.0098666,"\int (a+b \sec (e+f x))^m \, dx","Int[(a + b*Sec[e + f*x])^m,x]","\int (a+b \sec (e+f x))^m \, dx","\text{Int}\left((a+b \sec (e+f x))^m,x\right)",0,"Defer[Int][(a + b*Sec[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
792,0,0,0,0.0328224,"\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx","Int[Cos[e + f*x]*(a + b*Sec[e + f*x])^m,x]","\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\cos (e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Defer[Int][Cos[e + f*x]*(a + b*Sec[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
793,0,0,0,0.0406279,"\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx","Int[Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m,x]","\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\cos ^2(e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
794,1,135,0,0.1041059,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x]),x]","\frac{14 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{14 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 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{14 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{14 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 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,"(14*a*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*b*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (14*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,5,21,0.2381,1,"{4225, 2748, 2635, 2641, 2639}"
795,1,111,0,0.088547,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x]),x]","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(6*b*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*b*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",7,5,21,0.2381,1,"{4225, 2748, 2635, 2639, 2641}"
796,1,87,0,0.07644,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[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{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}","\frac{6 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 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,"(6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,5,21,0.2381,1,"{4225, 2748, 2635, 2641, 2639}"
797,1,61,0,0.066688,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[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{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)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*b*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",5,5,21,0.2381,1,"{4225, 2748, 2639, 2635, 2641}"
798,1,35,0,0.0573418,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[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)}{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)}{d}",1,"(2*a*EllipticE[(c + d*x)/2, 2])/d + (2*b*EllipticF[(c + d*x)/2, 2])/d","A",4,4,21,0.1905,1,"{4225, 2748, 2641, 2639}"
799,1,57,0,0.066769,"\int \frac{a+b \sec (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Sec[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 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)}{d}-\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-2*b*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/d + (2*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,21,0.2381,1,"{4225, 2748, 2636, 2639, 2641}"
800,1,83,0,0.0749682,"\int \frac{a+b \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[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)}{3 d}+\frac{2 b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d 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)}{3 d}+\frac{2 b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*a*EllipticE[(c + d*x)/2, 2])/d + (2*b*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,5,21,0.2381,1,"{4225, 2748, 2636, 2641, 2639}"
801,1,111,0,0.087413,"\int \frac{a+b \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[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{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 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{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-6*b*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*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*b*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,5,21,0.2381,1,"{4225, 2748, 2636, 2639, 2641}"
802,1,160,0,0.187469,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(7 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^2+9 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 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 \left(7 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^2+9 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 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}",1,"(2*(7*a^2 + 9*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*(7*a^2 + 9*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*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",10,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2641, 4045, 2639}"
803,1,135,0,0.172011,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(5 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 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 \left(5 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 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}",1,"(12*a*b*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a^2 + 7*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*a^2 + 7*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*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",9,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2639, 4045, 2641}"
804,1,101,0,0.1527978,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^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 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{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 \left(3 a^2+5 b^2\right) 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 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}",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) + (4*a*b*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",8,7,23,0.3043,1,"{4264, 3788, 3769, 3771, 2641, 4045, 2639}"
805,1,72,0,0.1390905,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^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) \sqrt{\cos (c+d x)}}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\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) \sqrt{\cos (c+d x)}}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",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*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",7,6,23,0.2609,1,"{4264, 3788, 3771, 2639, 4045, 2641}"
806,1,68,0,0.1361687,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}","\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(a^2 - b^2)*EllipticE[(c + d*x)/2, 2])/d + (4*a*b*EllipticF[(c + d*x)/2, 2])/d + (2*b^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,6,23,0.2609,1,"{4264, 3788, 3771, 2641, 4046, 2639}"
807,1,95,0,0.1448917,"\int \frac{(a+b \sec (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Sec[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{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d 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{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",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*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",8,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2639, 4046, 2641}"
808,1,135,0,0.1643168,"\int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^2/Cos[c + d*x]^(3/2),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{2 \left(5 a^2+3 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (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 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d 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{2 \left(5 a^2+3 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (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 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",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) + (2*b^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*(5*a^2 + 3*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",9,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2641, 4046, 2639}"
809,1,160,0,0.1826581,"\int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^2/Cos[c + d*x]^(5/2),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)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\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)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d 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)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\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)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",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*b^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (4*a*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (12*a*b*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",10,7,23,0.3043,1,"{4264, 3788, 3768, 3771, 2639, 4046, 2641}"
810,1,194,0,0.2864887,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3,x]","\frac{2 b \left(15 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(7 a^2+27 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{40 a^2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))}{9 d}","\frac{2 b \left(15 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(7 a^2+27 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{40 a^2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))}{9 d}",1,"(2*a*(7*a^2 + 27*b^2)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*b*(15*a^2 + 7*b^2)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(15*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*a^2 + 27*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a^2*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(9*d)","A",10,8,23,0.3478,1,"{4264, 3841, 4047, 3769, 3771, 2639, 4045, 2641}"
811,1,159,0,0.2602162,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3,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) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a^2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}","\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) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a^2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}",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) + (2*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a^2*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)","A",9,8,23,0.3478,1,"{4264, 3841, 4047, 3769, 3771, 2641, 4045, 2639}"
812,1,116,0,0.2279475,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3,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{8 a^2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}","\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{8 a^2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}",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*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)","A",8,7,23,0.3043,1,"{4264, 3841, 4047, 3771, 2639, 4045, 2641}"
813,1,126,0,0.2313603,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3,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 b \left(a^2-3 b^2\right) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}{3 d}","\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 b \left(a^2-3 b^2\right) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}{3 d}",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) - (2*b*(a^2 - 3*b^2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,23,0.3043,1,"{4264, 3841, 4047, 3771, 2641, 4046, 2639}"
814,1,118,0,0.2299155,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3 \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3,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{16 a b^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{3 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{16 a b^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{3 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) + (16*a*b^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",8,7,23,0.3043,1,"{4264, 3842, 4047, 3771, 2641, 4046, 2639}"
815,1,149,0,0.245708,"\int \frac{(a+b \sec (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Sec[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{6 b \left(5 a^2+b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \cos ^{\frac{3}{2}}(c+d 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{6 b \left(5 a^2+b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \cos ^{\frac{3}{2}}(c+d x)}",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*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (6*b*(5*a^2 + b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2))","A",9,8,23,0.3478,1,"{4264, 3842, 4047, 3768, 3771, 2639, 4046, 2641}"
816,1,194,0,0.2850381,"\int \frac{(a+b \sec (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^3/Cos[c + d*x]^(3/2),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)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{7 d \cos ^{\frac{5}{2}}(c+d 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)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{7 d \cos ^{\frac{5}{2}}(c+d x)}",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) + (32*a*b^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*b*(21*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*a^2 + 9*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2))","A",10,8,23,0.3478,1,"{4264, 3842, 4047, 3768, 3771, 2641, 4046, 2639}"
817,1,152,0,0.6022735,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x]),x]","-\frac{2 b \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d}+\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 b^4 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}","-\frac{2 b \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d}+\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 b^4 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"(2*(3*a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*b*(a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2])/(3*a^4*d) + (2*b^4*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)","A",11,10,23,0.4348,1,"{4264, 3853, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
818,1,112,0,0.3902382,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}-\frac{2 b^3 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}-\frac{2 b^3 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(-2*b*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2])/(3*a^3*d) - (2*b^3*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",10,9,23,0.3913,1,"{4264, 3853, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
819,1,75,0,0.2420275,"\int \frac{\sqrt{\cos (c+d x)}}{a+b \sec (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{2 b^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}","\frac{2 b^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(2*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*b*EllipticF[(c + d*x)/2, 2])/(a^2*d) + (2*b^2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d)","A",9,8,23,0.3478,1,"{4264, 3852, 3849, 2805, 3787, 3771, 2639, 2641}"
820,1,53,0,0.188849,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*EllipticF[(c + d*x)/2, 2])/(a*d) - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d)","A",5,5,23,0.2174,1,"{4264, 3848, 2803, 2641, 2805}"
821,1,29,0,0.1283315,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\frac{2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",3,3,23,0.1304,1,"{4264, 3849, 2805}"
822,1,77,0,0.2020382,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","-\frac{2 a \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}+\frac{2 \sin (c+d x)}{b d \sqrt{\cos (c+d x)}}","-\frac{2 a \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}+\frac{2 \sin (c+d x)}{b d \sqrt{\cos (c+d x)}}",1,"(-2*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) + (2*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",7,7,23,0.3043,1,"{4264, 3850, 3768, 3771, 2639, 3849, 2805}"
823,1,128,0,0.5404555,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]","\frac{2 a^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*a*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*EllipticF[(c + d*x)/2, 2])/(3*b*d) + (2*a^2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d) + (2*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) - (2*a*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])","A",11,10,23,0.4348,1,"{4264, 3851, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
824,1,244,0,0.7266556,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2,x]","\frac{\left(16 a^2 b^2+2 a^4-15 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{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^3 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}+\frac{\left(2 a^2-5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}","\frac{\left(16 a^2 b^2+2 a^4-15 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{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^3 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}+\frac{\left(2 a^2-5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (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^4 + 16*a^2*b^2 - 15*b^4)*EllipticF[(c + d*x)/2, 2])/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3847, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
825,1,184,0,0.4786776,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","-\frac{b \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 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^2 \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}","-\frac{b \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 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^2 \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"((2*a^2 - 3*b^2)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) - (b*(4*a^2 - 3*b^2)*EllipticF[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{4264, 3847, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
826,1,167,0,0.4194027,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\left(2 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{b \left(3 a^2-b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}","\frac{\left(2 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{b \left(3 a^2-b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"(b*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*EllipticF[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) - (b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{4264, 3843, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
827,1,148,0,0.3938736,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[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{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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (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{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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"-(EllipticE[(c + d*x)/2, 2]/((a^2 - b^2)*d)) - (b*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*(a + b)^2*d) + (a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{4264, 3844, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
828,1,154,0,0.4451692,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[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{a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{\left(a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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{\left(a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"(a*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + EllipticF[(c + d*x)/2, 2]/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a - b)*b*(a + b)^2*d) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,23,0.3913,1,"{4264, 3845, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
829,1,219,0,0.6840679,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[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{\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 \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (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{\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 \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}",1,"-(((3*a^2 - 2*b^2)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d)) - (a*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3845, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
830,1,346,0,1.0568716,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\left(128 a^4 b^2-223 a^2 b^4+8 a^6+105 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{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^3 \left(-86 a^2 b^2+63 a^4+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(13 a^2-7 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 \sec (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 \sec (c+d x))^2}","\frac{\left(128 a^4 b^2-223 a^2 b^4+8 a^6+105 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{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^3 \left(-86 a^2 b^2+63 a^4+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(-61 a^2 b^2+8 a^4+35 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(13 a^2-7 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 \sec (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 \sec (c+d x))^2}",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^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*EllipticF[(c + d*x)/2, 2])/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",12,11,23,0.4783,1,"{4264, 3847, 4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
831,1,282,0,0.8067879,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^3} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^3,x]","-\frac{3 b \left(-11 a^2 b^2+8 a^4+5 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 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^2 \left(-38 a^2 b^2+35 a^4+15 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b^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 \sec (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 \sec (c+d x))^2}","-\frac{3 b \left(-11 a^2 b^2+8 a^4+5 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-29 a^2 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^2 \left(-38 a^2 b^2+35 a^4+15 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b^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 \sec (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 \sec (c+d x))^2}",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) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*EllipticF[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[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*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3847, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
832,1,263,0,0.6828046,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\left(-5 a^2 b^2+8 a^4+3 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{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 b \left(-2 a^2 b^2+5 a^4+b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}-\frac{b \left(7 a^2-b^2\right) \sin (c+d x)}{4 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}-\frac{b \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}","\frac{\left(-5 a^2 b^2+8 a^4+3 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{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 b \left(-2 a^2 b^2+5 a^4+b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}-\frac{b \left(7 a^2-b^2\right) \sin (c+d x)}{4 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}-\frac{b \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (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) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*EllipticF[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) - (b*(7*a^2 - b^2)*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3843, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
833,1,246,0,0.6577352,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","-\frac{b \left(7 a^2-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(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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}+\frac{3 \left(a^2+b^2\right) \sin (c+d x)}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}","-\frac{b \left(7 a^2-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(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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}+\frac{3 \left(a^2+b^2\right) \sin (c+d x)}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}",1,"-((5*a^2 + b^2)*EllipticE[(c + d*x)/2, 2])/(4*a*(a^2 - b^2)^2*d) - (b*(7*a^2 - b^2)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*Sin[c + d*x])/(2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) + (3*(a^2 + b^2)*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3844, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
834,1,253,0,0.7214453,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[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 a 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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}+\frac{a \left(a^2-7 b^2\right) \sin (c+d x)}{4 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}","\frac{3 \left(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{\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 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}+\frac{a \left(a^2-7 b^2\right) \sin (c+d x)}{4 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"((a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(4*b*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2])/(4*a*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) + (a*(a^2 - 7*b^2)*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3845, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
835,1,255,0,0.7462592,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\left(a^2-7 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{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 \left(-2 a^2 b^2+a^4+5 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}","\frac{\left(a^2-7 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{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 \left(-2 a^2 b^2+a^4+5 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (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) + ((a^2 - 7*b^2)*EllipticF[(c + d*x)/2, 2])/(4*b*(a^2 - b^2)^2*d) + (3*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2) - (3*a^2*(a^2 - 3*b^2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,10,23,0.4348,1,"{4264, 3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
836,1,328,0,0.9998228,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Int[1/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3),x]","-\frac{a \left(5 a^2-11 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(-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 \left(-38 a^2 b^2+15 a^4+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\right) \sin (c+d x)}{4 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}","-\frac{a \left(5 a^2-11 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(-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 \left(-38 a^2 b^2+15 a^4+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}+\frac{\left(-29 a^2 b^2+15 a^4+8 b^4\right) \sin (c+d x)}{4 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}",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) - (a*(5*a^2 - 11*b^2)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) - (a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2) - (a^2*(5*a^2 - 11*b^2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))","A",12,11,23,0.4783,1,"{4264, 3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
837,1,244,0,0.6589072,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]],x]","-\frac{4 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a d}","-\frac{4 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a d}",1,"(-4*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,25,0.4000,1,"{4264, 3857, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
838,1,192,0,0.4320753,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Int[Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,25,0.3600,1,"{4264, 3857, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
839,1,67,0,0.1458621,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",4,4,25,0.1600,1,"{4264, 3856, 2655, 2653}"
840,1,138,0,0.4060168,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",8,8,25,0.3200,1,"{4264, 3854, 3858, 2663, 2661, 3859, 2807, 2805}"
841,1,237,0,0.6898466,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[Sqrt[a + b*Sec[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,25,0.5200,1,"{4264, 3855, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
842,1,303,0,0.9399513,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 a d}+\frac{2 \left(-31 a^2 b^2+25 a^4+6 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 d}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 a d}+\frac{2 \left(-31 a^2 b^2+25 a^4+6 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 d}",1,"(2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(41*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (16*b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",11,10,25,0.4000,1,"{4264, 3864, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
843,1,240,0,0.6814953,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 d}","\frac{2 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 d}",1,"(2*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(5*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (4*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,25,0.4000,1,"{4264, 3864, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
844,1,187,0,0.4691201,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (8*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,9,25,0.3600,1,"{4264, 3864, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
845,1,209,0,0.5517795,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",12,12,25,0.4800,1,"{4264, 3868, 3856, 2655, 2653, 3854, 3858, 2663, 2661, 3859, 2807, 2805}"
846,1,249,0,0.775235,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\frac{\left(2 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{\left(2 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"((2*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (3*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,25,0.5200,1,"{4264, 3866, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
847,1,299,0,1.0243798,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{7 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{7 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(7*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (5*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + (5*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])","A",14,14,25,0.5600,1,"{4264, 3866, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
848,1,363,0,1.3275658,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{4 b \left(-62 a^2 b^2+57 a^4+5 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 b^2+147 a^4-10 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{38 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{63 d}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{4 b \left(-62 a^2 b^2+57 a^4+5 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(279 a^2 b^2+147 a^4-10 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{38 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{63 d}",1,"(4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(163*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2 + 75*b^2)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (38*a*b*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",12,10,25,0.4000,1,"{4264, 3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
849,1,303,0,1.0058138,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 \left(-2 a^2 b^2+5 a^4-3 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{6 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 \left(-2 a^2 b^2+5 a^4-3 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{6 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(21*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(29*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(21*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (6*a*b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",11,10,25,0.4000,1,"{4264, 3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
850,1,239,0,0.7557033,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{16 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{22 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}","\frac{16 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{22 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}",1,"(16*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 23*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (22*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,25,0.4000,1,"{4264, 3841, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
851,1,262,0,0.8527412,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b^3 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b^3 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*a*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (14*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",13,13,25,0.5200,1,"{4264, 3841, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
852,1,263,0,0.8490443,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{b \left(4 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{5 a b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{b \left(4 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{5 a b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(b*(4*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (5*a*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,25,0.5200,1,"{4264, 3842, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
853,1,314,0,1.1519924,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(a*(8*a^2 + 11*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (9*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + (9*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])","A",14,14,25,0.5600,1,"{4264, 3842, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
854,1,369,0,1.420618,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(b*(59*a^2 + 16*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (5*a*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((33*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + (13*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Cos[c + d*x]^(3/2)) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]])","A",15,14,25,0.5600,1,"{4264, 3842, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
855,1,249,0,0.6513089,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^(5/2)/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{8 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}","-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{8 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(-2*b*(7*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 8*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (8*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)","A",10,10,25,0.4000,1,"{4264, 3863, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
856,1,195,0,0.4392673,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Cos[c + d*x]^(3/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a d}","\frac{2 \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"(2*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (4*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",9,9,25,0.3600,1,"{4264, 3863, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
857,1,142,0,0.3072142,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[Sqrt[Cos[c + d*x]]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",8,8,25,0.3200,1,"{4264, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
858,1,67,0,0.1492713,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{4264, 3858, 2663, 2661}"
859,1,68,0,0.2347621,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",4,4,25,0.1600,1,"{4264, 3859, 2807, 2805}"
860,1,246,0,0.6962522,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",13,13,25,0.5200,1,"{4264, 3860, 4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
861,1,312,0,0.9522219,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"-(a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (3*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) - (3*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])","A",14,14,25,0.5600,1,"{4264, 3860, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
862,1,360,0,1.050346,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(8 a^2 b^2+3 a^4-16 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-8*b*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(5*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(3*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d) + (2*(a^2 - 6*b^2)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)","A",11,10,25,0.4000,1,"{4264, 3847, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
863,1,289,0,0.7523642,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(5*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)","A",10,10,25,0.4000,1,"{4264, 3847, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
864,1,214,0,0.5255194,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Sec[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 \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(-4*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{4264, 3847, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
865,1,200,0,0.4621873,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[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 \sec (c+d x)}}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",9,9,25,0.3600,1,"{4264, 3843, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
866,1,126,0,0.2331275,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[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 \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/((a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",6,6,25,0.2400,1,"{4264, 3844, 21, 3856, 2655, 2653}"
867,1,206,0,0.5982322,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[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 \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,25,0.4000,1,"{4264, 3845, 4108, 3859, 2807, 2805, 21, 3856, 2655, 2653}"
868,1,345,0,1.0888885,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]","-\frac{2 a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (3*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",14,14,25,0.5600,1,"{4264, 3845, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
869,1,391,0,1.0954161,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{4 b^2 \left(5 a^2-3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-13 a^2 b^2+a^4+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(16 a^2 b^2+a^4-16 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{4 b^2 \left(5 a^2-3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-13 a^2 b^2+a^4+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(16 a^2 b^2+a^4-16 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(a^4 + 16*a^2*b^2 - 16*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",11,11,25,0.4400,1,"{4264, 3847, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
870,1,317,0,0.822351,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[Sqrt[Cos[c + d*x]]/(a + b*Sec[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 \sec (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 \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{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 \sec (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 \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-15 a^2 b^2+3 a^4+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*b*(9*a^2 - 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[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*Sec[c + d*x]])","A",10,10,25,0.4000,1,"{4264, 3847, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
871,1,302,0,0.7400316,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","-\frac{2 b \left(5 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 \sec (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","-\frac{2 b \left(5 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 \sec (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(3*a^2 - 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,25,0.4000,1,"{4264, 3843, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
872,1,281,0,0.6879856,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{4 \left(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 \sec (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{4 \left(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 \sec (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*(a^2 + b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,25,0.4000,1,"{4264, 3844, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
873,1,277,0,0.7445737,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (8*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(a^2 - 5*b^2)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,25,0.4000,1,"{4264, 3845, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
874,1,370,0,1.2046757,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Int[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)),x]","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^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 \sec (c+d x)}}-\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^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 \sec (c+d x)}}-\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(-2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*(3*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*a^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*Sec[c + d*x]])","A",14,14,25,0.5600,1,"{4264, 3845, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
875,1,266,0,0.4562454,"\int (d \cos (e+f x))^n (a+b \sec (e+f x))^3 \, dx","Int[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^3,x]","-\frac{b \left(3 a^2 (2-n)+b^2 (1-n)\right) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a \left(a^2 (1-n)-3 b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \cos (e+f x))^n}{f (2-n)}","-\frac{b \left(3 a^2 (2-n)+b^2 (1-n)\right) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a \left(a^2 (1-n)-3 b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \cos (e+f x))^n}{f (2-n)}",1,"-((b*(b^2*(1 - n) + 3*a^2*(2 - n))*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a*(a^2*(1 - n) - 3*b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + (b^2*(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n))","A",8,6,23,0.2609,1,"{4264, 3842, 4047, 3772, 2643, 4046}"
876,1,186,0,0.2239338,"\int (d \cos (e+f x))^n (a+b \sec (e+f x))^2 \, dx","Int[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^2,x]","-\frac{\left(a^2 (1-n)-b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{2 a b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}","-\frac{\left(a^2 (1-n)-b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{2 a b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}",1,"(-2*a*b*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2]) - ((a^2*(1 - n) - b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (b^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n))","A",7,5,23,0.2174,1,"{4264, 3788, 3772, 2643, 4046}"
877,1,132,0,0.1149971,"\int (d \cos (e+f x))^n (a+b \sec (e+f x)) \, dx","Int[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x]),x]","-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}","-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}",1,"-((b*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*(1 + n)*Sqrt[Sin[e + f*x]^2])","A",5,4,21,0.1905,1,"{4225, 16, 2748, 2643}"
878,1,196,0,0.3654247,"\int \frac{(d \cos (e+f x))^n}{a+b \sec (e+f x)} \, dx","Int[(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};-\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};-\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"(a*AppellF1[1/2, (-1 - n)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((-1 - n)/2)*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, -n/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(d*Cos[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f*(Cos[e + f*x]^2)^(n/2))","A",7,5,23,0.2174,1,"{4264, 3869, 2823, 3189, 429}"
879,1,309,0,0.5139056,"\int \frac{(d \cos (e+f x))^n}{(a+b \sec (e+f x))^2} \, dx","Int[(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x])^2,x]","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"(a^2*AppellF1[1/2, (-3 - n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((-1 - n)/2)*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (-1 - n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((-1 - n)/2)*Sin[e + f*x])/((a^2 - b^2)^2*f) - (2*a*b*AppellF1[1/2, (-2 - n)/2, 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(d*Cos[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f*(Cos[e + f*x]^2)^(n/2))","A",10,5,23,0.2174,1,"{4264, 3869, 2824, 3189, 429}"