1,1,152,0,0.1071193,"\int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^9,x]","-\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^8(c+d x)}{8 d}+\frac{4 a \cos ^7(c+d x)}{7 d}+\frac{2 a \cos ^6(c+d x)}{3 d}-\frac{6 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{2 d}+\frac{4 a \cos ^3(c+d x)}{3 d}+\frac{2 a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^8(c+d x)}{8 d}+\frac{4 a \cos ^7(c+d x)}{7 d}+\frac{2 a \cos ^6(c+d x)}{3 d}-\frac{6 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{2 d}+\frac{4 a \cos ^3(c+d x)}{3 d}+\frac{2 a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (2*a*Cos[c + d*x]^2)/d + (4*a*Cos[c + d*x]^3)/(3*d) - (3*a*Cos[c + d*x]^4)/(2*d) - (6*a*Cos[c + d*x]^5)/(5*d) + (2*a*Cos[c + d*x]^6)/(3*d) + (4*a*Cos[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]^8)/(8*d) - (a*Cos[c + d*x]^9)/(9*d) - (a*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
2,1,119,0,0.0974775,"\int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^7,x]","\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^6(c+d x)}{6 d}-\frac{3 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{4 d}+\frac{a \cos ^3(c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^6(c+d x)}{6 d}-\frac{3 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{4 d}+\frac{a \cos ^3(c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (3*a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*a*Cos[c + d*x]^4)/(4*d) - (3*a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (a*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
3,1,87,0,0.0869581,"\int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^5,x]","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^4(c+d x)}{4 d}+\frac{2 a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^4(c+d x)}{4 d}+\frac{2 a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (a*Cos[c + d*x]^2)/d + (2*a*Cos[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]^4)/(4*d) - (a*Cos[c + d*x]^5)/(5*d) - (a*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
4,1,58,0,0.0773241,"\int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^3,x]","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/(3*d) - (a*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2836, 12, 75}"
5,1,26,0,0.0306214,"\int (a+a \sec (c+d x)) \sin (c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x],x]","-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) - (a*Log[Cos[c + d*x]])/d","A",4,3,17,0.1765,1,"{3872, 2707, 43}"
6,1,30,0,0.058121,"\int \csc (c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{a \log (1-\cos (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{a \log (1-\cos (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(a*Log[1 - Cos[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d","A",6,6,17,0.3529,1,"{3872, 2836, 12, 36, 31, 29}"
7,1,73,0,0.0954302,"\int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^3*(a + a*Sec[c + d*x]),x]","-\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (\cos (c+d x)+1)}{4 d}","-\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (\cos (c+d x)+1)}{4 d}",1,"-a^2/(2*d*(a - a*Cos[c + d*x])) + (3*a*Log[1 - Cos[c + d*x]])/(4*d) - (a*Log[Cos[c + d*x]])/d + (a*Log[1 + Cos[c + d*x]])/(4*d)","A",5,4,19,0.2105,1,"{3872, 2836, 12, 72}"
8,1,118,0,0.120413,"\int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x]),x]","-\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a \cos (c+d x)+a)}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}","-\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a \cos (c+d x)+a)}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}",1,"-a^3/(8*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^2/(8*d*(a + a*Cos[c + d*x])) + (11*a*Log[1 - Cos[c + d*x]])/(16*d) - (a*Log[Cos[c + d*x]])/d + (5*a*Log[1 + Cos[c + d*x]])/(16*d)","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
9,1,163,0,0.149683,"\int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x]),x]","-\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^3}{32 d (a \cos (c+d x)+a)^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{3 a^2}{16 d (a \cos (c+d x)+a)}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (\cos (c+d x)+1)}{32 d}","-\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^3}{32 d (a \cos (c+d x)+a)^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{3 a^2}{16 d (a \cos (c+d x)+a)}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (\cos (c+d x)+1)}{32 d}",1,"-a^4/(24*d*(a - a*Cos[c + d*x])^3) - (5*a^3)/(32*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^3/(32*d*(a + a*Cos[c + d*x])^2) - (3*a^2)/(16*d*(a + a*Cos[c + d*x])) + (21*a*Log[1 - Cos[c + d*x]])/(32*d) - (a*Log[Cos[c + d*x]])/d + (11*a*Log[1 + Cos[c + d*x]])/(32*d)","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
10,1,165,0,0.1456376,"\int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^8,x]","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^7(c+d x) \cos (c+d x)}{8 d}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{48 d}-\frac{35 a \sin ^3(c+d x) \cos (c+d x)}{192 d}-\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128}","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^7(c+d x) \cos (c+d x)}{8 d}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{48 d}-\frac{35 a \sin ^3(c+d x) \cos (c+d x)}{192 d}-\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128}",1,"(35*a*x)/128 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a*Sin[c + d*x]^3)/(3*d) - (35*a*Cos[c + d*x]*Sin[c + d*x]^3)/(192*d) - (a*Sin[c + d*x]^5)/(5*d) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(48*d) - (a*Sin[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]*Sin[c + d*x]^7)/(8*d)","A",11,7,19,0.3684,1,"{3872, 2838, 2592, 302, 206, 2635, 8}"
11,1,127,0,0.1278329,"\int (a+a \sec (c+d x)) \sin ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^6,x]","-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}","-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(5*a*x)/16 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Sin[c + d*x]^3)/(3*d) - (5*a*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",10,7,19,0.3684,1,"{3872, 2838, 2592, 302, 206, 2635, 8}"
12,1,89,0,0.111148,"\int (a+a \sec (c+d x)) \sin ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^4,x]","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^3(c+d x) \cos (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 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^3(c+d x) \cos (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*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Sin[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",9,7,19,0.3684,1,"{3872, 2838, 2592, 302, 206, 2635, 8}"
13,1,51,0,0.0818246,"\int (a+a \sec (c+d x)) \sin ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])*Sin[c + d*x]^2,x]","-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\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 \tanh ^{-1}(\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*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",7,7,19,0.3684,1,"{3872, 2838, 2592, 321, 206, 2635, 8}"
14,1,37,0,0.0934906,"\int \csc ^2(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^2*(a + a*Sec[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Csc[c + d*x])/d","A",7,7,19,0.3684,1,"{3872, 2838, 2621, 321, 207, 3767, 8}"
15,1,69,0,0.1025162,"\int \csc ^4(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^4*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
16,1,101,0,0.1095021,"\int \csc ^6(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^6*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (2*a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
17,1,131,0,0.1170039,"\int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^8*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{3 a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{3 a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/d - (3*a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
18,1,165,0,0.1266022,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx","Int[Csc[c + d*x]^10*(a + a*Sec[c + d*x]),x]","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{4 a \cot ^7(c+d x)}{7 d}-\frac{6 a \cot ^5(c+d x)}{5 d}-\frac{4 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{4 a \cot ^7(c+d x)}{7 d}-\frac{6 a \cot ^5(c+d x)}{5 d}-\frac{4 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (4*a*Cot[c + d*x]^3)/(3*d) - (6*a*Cot[c + d*x]^5)/(5*d) - (4*a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
19,1,183,0,0.1877935,"\int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^9,x]","-\frac{a^2 \cos ^9(c+d x)}{9 d}-\frac{a^2 \cos ^8(c+d x)}{4 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{4 a^2 \cos ^2(c+d x)}{d}+\frac{3 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","-\frac{a^2 \cos ^9(c+d x)}{9 d}-\frac{a^2 \cos ^8(c+d x)}{4 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{4 a^2 \cos ^2(c+d x)}{d}+\frac{3 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(3*a^2*Cos[c + d*x])/d + (4*a^2*Cos[c + d*x]^2)/d - (2*a^2*Cos[c + d*x]^3)/(3*d) - (3*a^2*Cos[c + d*x]^4)/d - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^6)/(3*d) + (3*a^2*Cos[c + d*x]^7)/(7*d) - (a^2*Cos[c + d*x]^8)/(4*d) - (a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
20,1,131,0,0.168269,"\int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^7,x]","\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{2 d}+\frac{3 a^2 \cos ^2(c+d x)}{d}+\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{2 d}+\frac{3 a^2 \cos ^2(c+d x)}{d}+\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(2*a^2*Cos[c + d*x])/d + (3*a^2*Cos[c + d*x]^2)/d - (3*a^2*Cos[c + d*x]^4)/(2*d) - (2*a^2*Cos[c + d*x]^5)/(5*d) + (a^2*Cos[c + d*x]^6)/(3*d) + (a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
21,1,112,0,0.1576341,"\int (a+a \sec (c+d x))^2 \sin ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^5,x]","-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \cos ^4(c+d x)}{2 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \cos ^4(c+d x)}{2 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cos[c + d*x]^4)/(2*d) - (a^2*Cos[c + d*x]^5)/(5*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
22,1,62,0,0.1235906,"\int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^3,x]","\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 75}"
23,1,43,0,0.0767932,"\int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x],x]","-\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","-\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"-((a^2*Cos[c + d*x])/d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
24,1,48,0,0.1151482,"\int \csc (c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(2*a^2*Log[1 - Cos[c + d*x]])/d - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,19,0.2105,1,"{3872, 2836, 12, 77}"
25,1,69,0,0.1440592,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","-\frac{a^3}{d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}","-\frac{a^3}{d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"-(a^3/(d*(a - a*Cos[c + d*x]))) + (2*a^2*Log[1 - Cos[c + d*x]])/d - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 44}"
26,1,115,0,0.1712221,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]","-\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{8 d}","-\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{8 d}",1,"-a^4/(4*d*(a - a*Cos[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Cos[c + d*x])) + (17*a^2*Log[1 - Cos[c + d*x]])/(8*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(8*d) + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
27,1,160,0,0.1992068,"\int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]","-\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{4 d}","-\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{4 d}",1,"-a^5/(12*d*(a - a*Cos[c + d*x])^3) - (3*a^4)/(8*d*(a - a*Cos[c + d*x])^2) - (23*a^3)/(16*d*(a - a*Cos[c + d*x])) + a^3/(16*d*(a + a*Cos[c + d*x])) + (9*a^2*Log[1 - Cos[c + d*x]])/(4*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(4*d) + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
28,1,205,0,0.238155,"\int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^2,x]","-\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac{a^4}{64 d (a \cos (c+d x)+a)^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{9 a^3}{64 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (\cos (c+d x)+1)}{128 d}","-\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac{a^4}{64 d (a \cos (c+d x)+a)^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{9 a^3}{64 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (\cos (c+d x)+1)}{128 d}",1,"-a^6/(32*d*(a - a*Cos[c + d*x])^4) - (7*a^5)/(48*d*(a - a*Cos[c + d*x])^3) - (15*a^4)/(32*d*(a - a*Cos[c + d*x])^2) - (51*a^3)/(32*d*(a - a*Cos[c + d*x])) + a^4/(64*d*(a + a*Cos[c + d*x])^2) + (9*a^3)/(64*d*(a + a*Cos[c + d*x])) + (303*a^2*Log[1 - Cos[c + d*x]])/(128*d) - (2*a^2*Log[Cos[c + d*x]])/d - (47*a^2*Log[1 + Cos[c + d*x]])/(128*d) + (a^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
29,1,199,0,0.3607794,"\int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^8,x]","-\frac{2 a^2 \sin ^7(c+d x)}{7 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{245 a^2 x}{128}","-\frac{2 a^2 \sin ^7(c+d x)}{7 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{245 a^2 x}{128}",1,"(-245*a^2*x)/128 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d + (139*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (17*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (2*a^2*Sin[c + d*x]^3)/(3*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) - (2*a^2*Sin[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d","A",27,8,21,0.3810,1,"{3872, 2872, 2637, 2635, 8, 2633, 3770, 3767}"
30,1,157,0,0.2699592,"\int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^6,x]","-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{25 a^2 x}{16}","-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{25 a^2 x}{16}",1,"(-25*a^2*x)/16 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (7*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a^2*Sin[c + d*x]^3)/(3*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) + (a^2*Tan[c + d*x])/d","A",18,8,21,0.3810,1,"{3872, 2872, 2637, 2633, 2635, 8, 3770, 3767}"
31,1,115,0,0.2684334,"\int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^4,x]","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{9 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 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{9 a^2 x}{8}",1,"(-9*a^2*x)/8 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (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^2*Tan[c + d*x])/d","A",14,8,21,0.3810,1,"{3872, 2872, 2637, 2635, 8, 2633, 3770, 3767}"
32,1,73,0,0.1321852,"\int (a+a \sec (c+d x))^2 \sin ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^2,x]","-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a^2 x}{2}","-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a^2 x}{2}",1,"-(a^2*x)/2 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*Tan[c + d*x])/d","A",9,7,21,0.3333,1,"{3872, 2709, 2637, 2635, 8, 3770, 3767}"
33,1,57,0,0.2464873,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Cot[c + d*x])/d - (2*a^2*Csc[c + d*x])/d + (a^2*Tan[c + d*x])/d","A",11,9,21,0.4286,1,"{3872, 2873, 3767, 8, 2621, 321, 207, 2620, 14}"
34,1,87,0,0.2974412,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","\frac{10 a^2 \tan (c+d x)}{3 d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}-\frac{2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}","\frac{10 a^2 \tan (c+d x)}{3 d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}-\frac{2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/d + (10*a^2*Tan[c + d*x])/(3*d) - (2*a^2*Tan[c + d*x])/(d*(1 - Cos[c + d*x])) - (a^4*Tan[c + d*x])/(3*d*(a - a*Cos[c + d*x])^2)","A",8,8,21,0.3810,1,"{3872, 2869, 2766, 2978, 2748, 3767, 8, 3770}"
35,1,129,0,0.2263375,"\int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^6*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{5 a^2 \cot ^3(c+d x)}{3 d}-\frac{4 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{5 a^2 \cot ^3(c+d x)}{3 d}-\frac{4 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/d - (4*a^2*Cot[c + d*x])/d - (5*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) + (a^2*Tan[c + d*x])/d","A",12,8,21,0.3810,1,"{3872, 2873, 3767, 2621, 302, 207, 2620, 270}"
36,1,163,0,0.2427918,"\int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{7 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \cot ^3(c+d x)}{d}-\frac{5 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{7 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \cot ^3(c+d x)}{d}-\frac{5 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/d - (5*a^2*Cot[c + d*x])/d - (3*a^2*Cot[c + d*x]^3)/d - (7*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d","A",12,8,21,0.3810,1,"{3872, 2873, 3767, 2621, 302, 207, 2620, 270}"
37,1,201,0,0.2584129,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{9 a^2 \cot ^7(c+d x)}{7 d}-\frac{16 a^2 \cot ^5(c+d x)}{5 d}-\frac{14 a^2 \cot ^3(c+d x)}{3 d}-\frac{6 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{9 a^2 \cot ^7(c+d x)}{7 d}-\frac{16 a^2 \cot ^5(c+d x)}{5 d}-\frac{14 a^2 \cot ^3(c+d x)}{3 d}-\frac{6 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/d - (6*a^2*Cot[c + d*x])/d - (14*a^2*Cot[c + d*x]^3)/(3*d) - (16*a^2*Cot[c + d*x]^5)/(5*d) - (9*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d) + (a^2*Tan[c + d*x])/d","A",12,8,21,0.3810,1,"{3872, 2873, 3767, 2621, 302, 207, 2620, 270}"
38,1,203,0,0.1956329,"\int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^9,x]","-\frac{a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}","-\frac{a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"(11*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (14*a^3*Cos[c + d*x]^3)/(3*d) - (7*a^3*Cos[c + d*x]^4)/(2*d) + (6*a^3*Cos[c + d*x]^5)/(5*d) + (11*a^3*Cos[c + d*x]^6)/(6*d) + (a^3*Cos[c + d*x]^7)/(7*d) - (3*a^3*Cos[c + d*x]^8)/(8*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
39,1,131,0,0.1681603,"\int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^7,x]","\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^6(c+d x)}{2 d}-\frac{2 a^3 \cos ^4(c+d x)}{d}-\frac{2 a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{8 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}","\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^6(c+d x)}{2 d}-\frac{2 a^3 \cos ^4(c+d x)}{d}-\frac{2 a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{8 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}",1,"(8*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (2*a^3*Cos[c + d*x]^3)/d - (2*a^3*Cos[c + d*x]^4)/d + (a^3*Cos[c + d*x]^6)/(2*d) + (a^3*Cos[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
40,1,134,0,0.1667556,"\int (a+a \sec (c+d x))^3 \sin ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^5,x]","-\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^4(c+d x)}{4 d}-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos ^2(c+d x)}{2 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}","-\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^4(c+d x)}{4 d}-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos ^2(c+d x)}{2 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"(5*a^3*Cos[c + d*x])/d + (5*a^3*Cos[c + d*x]^2)/(2*d) - (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]^4)/(4*d) - (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
41,1,98,0,0.0959318,"\int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^3,x]","\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{2 a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{2 a^3 \log (\cos (c+d x))}{d}",1,"(2*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",4,3,21,0.1429,1,"{3872, 2707, 75}"
42,1,62,0,0.0914808,"\int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x],x]","-\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}","-\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}",1,"-((a^3*Cos[c + d*x])/d) - (3*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
43,1,67,0,0.125251,"\int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}","\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}",1,"(4*a^3*Log[1 - Cos[c + d*x]])/d - (4*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,19,0.2105,1,"{3872, 2836, 12, 88}"
44,1,88,0,0.1563468,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*(a + a*Sec[c + d*x])^3,x]","-\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d}","-\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d}",1,"(-2*a^4)/(d*(a - a*Cos[c + d*x])) + (5*a^3*Log[1 - Cos[c + d*x]])/d - (5*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 77}"
45,1,111,0,0.1688264,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]","-\frac{a^5}{2 d (a-a \cos (c+d x))^2}-\frac{3 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{6 a^3 \log (1-\cos (c+d x))}{d}-\frac{6 a^3 \log (\cos (c+d x))}{d}","-\frac{a^5}{2 d (a-a \cos (c+d x))^2}-\frac{3 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{6 a^3 \log (1-\cos (c+d x))}{d}-\frac{6 a^3 \log (\cos (c+d x))}{d}",1,"-a^5/(2*d*(a - a*Cos[c + d*x])^2) - (3*a^4)/(d*(a - a*Cos[c + d*x])) + (6*a^3*Log[1 - Cos[c + d*x]])/d - (6*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 44}"
46,1,157,0,0.1952684,"\int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]","-\frac{a^6}{6 d (a-a \cos (c+d x))^3}-\frac{7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac{31 a^4}{8 d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{7 a^3 \log (\cos (c+d x))}{d}+\frac{a^3 \log (\cos (c+d x)+1)}{16 d}","-\frac{a^6}{6 d (a-a \cos (c+d x))^3}-\frac{7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac{31 a^4}{8 d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{7 a^3 \log (\cos (c+d x))}{d}+\frac{a^3 \log (\cos (c+d x)+1)}{16 d}",1,"-a^6/(6*d*(a - a*Cos[c + d*x])^3) - (7*a^5)/(8*d*(a - a*Cos[c + d*x])^2) - (31*a^4)/(8*d*(a - a*Cos[c + d*x])) + (111*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (7*a^3*Log[Cos[c + d*x]])/d + (a^3*Log[1 + Cos[c + d*x]])/(16*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
47,1,202,0,0.2305948,"\int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]","-\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a \cos (c+d x)+a)}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (\cos (c+d x)+1)}{64 d}","-\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a \cos (c+d x)+a)}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (\cos (c+d x)+1)}{64 d}",1,"-a^7/(16*d*(a - a*Cos[c + d*x])^4) - a^6/(3*d*(a - a*Cos[c + d*x])^3) - (39*a^5)/(32*d*(a - a*Cos[c + d*x])^2) - (75*a^4)/(16*d*(a - a*Cos[c + d*x])) - a^4/(32*d*(a + a*Cos[c + d*x])) + (501*a^3*Log[1 - Cos[c + d*x]])/(64*d) - (8*a^3*Log[Cos[c + d*x]])/d + (11*a^3*Log[1 + Cos[c + d*x]])/(64*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
48,1,210,0,0.389447,"\int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^8,x]","-\frac{3 a^3 \sin ^7(c+d x)}{7 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac{293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{805 a^3 x}{128}","-\frac{3 a^3 \sin ^7(c+d x)}{7 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac{293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{805 a^3 x}{128}",1,"(-805*a^3*x)/128 - (a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a^3*Sin[c + d*x]^3)/(3*d) - (2*a^3*Sin[c + d*x]^5)/(5*d) - (3*a^3*Sin[c + d*x]^7)/(7*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",29,9,21,0.4286,1,"{3872, 2872, 2637, 2635, 8, 2633, 3770, 3767, 3768}"
49,1,182,0,0.2736806,"\int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^6,x]","-\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{2 a^3 \sin ^3(c+d x)}{3 d}-\frac{a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{85 a^3 x}{16}","-\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{2 a^3 \sin ^3(c+d x)}{3 d}-\frac{a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{85 a^3 x}{16}",1,"(-85*a^3*x)/16 + (a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Sin[c + d*x])/d + (43*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*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) - (2*a^3*Sin[c + d*x]^3)/(3*d) - (3*a^3*Sin[c + d*x]^5)/(5*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",18,9,21,0.4286,1,"{3872, 2872, 2637, 2635, 8, 2633, 3767, 3768, 3770}"
50,1,138,0,0.2276646,"\int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^4,x]","-\frac{a^3 \sin ^3(c+d x)}{d}-\frac{2 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{33 a^3 x}{8}","-\frac{a^3 \sin ^3(c+d x)}{d}-\frac{2 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{33 a^3 x}{8}",1,"(-33*a^3*x)/8 + (3*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (2*a^3*Sin[c + d*x])/d + (7*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 + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",16,9,21,0.4286,1,"{3872, 2872, 2637, 2635, 8, 2633, 3770, 3767, 3768}"
51,1,98,0,0.1831769,"\int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^2,x]","-\frac{3 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 x}{2}","-\frac{3 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 x}{2}",1,"(-5*a^3*x)/2 + (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^3*Sin[c + d*x])/d - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,8,21,0.3810,1,"{3872, 2872, 2637, 2635, 8, 3770, 3767, 3768}"
52,1,80,0,0.1940483,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]","\frac{3 a^3 \tan (c+d x)}{d}+\frac{9 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{4 a^3 \sin (c+d x)}{d (1-\cos (c+d x))}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{3 a^3 \tan (c+d x)}{d}+\frac{9 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{4 a^3 \sin (c+d x)}{d (1-\cos (c+d x))}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(9*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (4*a^3*Sin[c + d*x])/(d*(1 - Cos[c + d*x])) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",9,7,21,0.3333,1,"{3872, 2872, 2648, 3770, 3767, 8, 3768}"
53,1,110,0,0.229634,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","\frac{3 a^3 \tan (c+d x)}{d}+\frac{11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac{2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{3 a^3 \tan (c+d x)}{d}+\frac{11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac{2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(11*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (2*a^3*Sin[c + d*x])/(3*d*(1 - Cos[c + d*x])^2) - (17*a^3*Sin[c + d*x])/(3*d*(1 - Cos[c + d*x])) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",11,8,21,0.3810,1,"{3872, 2872, 2650, 2648, 3770, 3767, 8, 3768}"
54,1,165,0,0.4356165,"\int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^6*(a + a*Sec[c + d*x])^3,x]","\frac{152 a^3 \tan (c+d x)}{15 d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3-a^3 \cos (c+d x)\right)}-\frac{a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac{11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2}","\frac{152 a^3 \tan (c+d x)}{15 d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3-a^3 \cos (c+d x)\right)}-\frac{a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac{11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2}",1,"(13*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (152*a^3*Tan[c + d*x])/(15*d) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (a^6*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a - a*Cos[c + d*x])^3) - (11*a^5*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a - a*Cos[c + d*x])^2) - (76*a^6*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 - a^3*Cos[c + d*x]))","A",10,9,21,0.4286,1,"{3872, 2869, 2766, 2978, 2748, 3768, 3770, 3767, 8}"
55,1,192,0,0.3143722,"\int \csc ^8(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{15 a^3 \csc (c+d x)}{2 d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{15 a^3 \csc (c+d x)}{2 d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}",1,"(15*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (13*a^3*Cot[c + d*x])/d - (7*a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/d - (4*a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Csc[c + d*x])/(2*d) - (5*a^3*Csc[c + d*x]^3)/(2*d) - (3*a^3*Csc[c + d*x]^5)/(2*d) - (15*a^3*Csc[c + d*x]^7)/(14*d) + (a^3*Csc[c + d*x]^7*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d","A",17,9,21,0.4286,1,"{3872, 2873, 3767, 2621, 302, 207, 2620, 270, 288}"
56,1,232,0,0.3316661,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{19 a^3 \cot ^7(c+d x)}{7 d}-\frac{36 a^3 \cot ^5(c+d x)}{5 d}-\frac{34 a^3 \cot ^3(c+d x)}{3 d}-\frac{16 a^3 \cot (c+d x)}{d}-\frac{17 a^3 \csc ^9(c+d x)}{18 d}-\frac{17 a^3 \csc ^7(c+d x)}{14 d}-\frac{17 a^3 \csc ^5(c+d x)}{10 d}-\frac{17 a^3 \csc ^3(c+d x)}{6 d}-\frac{17 a^3 \csc (c+d x)}{2 d}+\frac{17 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{19 a^3 \cot ^7(c+d x)}{7 d}-\frac{36 a^3 \cot ^5(c+d x)}{5 d}-\frac{34 a^3 \cot ^3(c+d x)}{3 d}-\frac{16 a^3 \cot (c+d x)}{d}-\frac{17 a^3 \csc ^9(c+d x)}{18 d}-\frac{17 a^3 \csc ^7(c+d x)}{14 d}-\frac{17 a^3 \csc ^5(c+d x)}{10 d}-\frac{17 a^3 \csc ^3(c+d x)}{6 d}-\frac{17 a^3 \csc (c+d x)}{2 d}+\frac{17 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}",1,"(17*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (16*a^3*Cot[c + d*x])/d - (34*a^3*Cot[c + d*x]^3)/(3*d) - (36*a^3*Cot[c + d*x]^5)/(5*d) - (19*a^3*Cot[c + d*x]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Csc[c + d*x])/(2*d) - (17*a^3*Csc[c + d*x]^3)/(6*d) - (17*a^3*Csc[c + d*x]^5)/(10*d) - (17*a^3*Csc[c + d*x]^7)/(14*d) - (17*a^3*Csc[c + d*x]^9)/(18*d) + (a^3*Csc[c + d*x]^9*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d","A",17,9,21,0.4286,1,"{3872, 2873, 3767, 2621, 302, 207, 2620, 270, 288}"
57,1,91,0,0.1612024,"\int \frac{\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x]),x]","\frac{\sin ^8(c+d x)}{8 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}","\frac{\sin ^8(c+d x)}{8 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]^5)/(5*a*d) + (3*Cos[c + d*x]^7)/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + Sin[c + d*x]^8/(8*a*d)","A",7,6,21,0.2857,1,"{3872, 2835, 2564, 30, 2565, 270}"
58,1,73,0,0.1547613,"\int \frac{\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^7/(a + a*Sec[c + d*x]),x]","\frac{\sin ^6(c+d x)}{6 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}","\frac{\sin ^6(c+d x)}{6 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"Cos[c + d*x]^3/(3*a*d) - (2*Cos[c + d*x]^5)/(5*a*d) + Cos[c + d*x]^7/(7*a*d) + Sin[c + d*x]^6/(6*a*d)","A",7,6,21,0.2857,1,"{3872, 2835, 2564, 30, 2565, 270}"
59,1,55,0,0.1481235,"\int \frac{\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^5/(a + a*Sec[c + d*x]),x]","\frac{\sin ^4(c+d x)}{4 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}","\frac{\sin ^4(c+d x)}{4 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) + Sin[c + d*x]^4/(4*a*d)","A",7,6,21,0.2857,1,"{3872, 2835, 2564, 30, 2565, 14}"
60,1,37,0,0.1261939,"\int \frac{\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{\sin ^2(c+d x)}{2 a d}+\frac{\cos ^3(c+d x)}{3 a d}","\frac{\sin ^2(c+d x)}{2 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"Cos[c + d*x]^3/(3*a*d) + Sin[c + d*x]^2/(2*a*d)","A",6,5,21,0.2381,1,"{3872, 2835, 2564, 30, 2565}"
61,1,31,0,0.0714901,"\int \frac{\sin (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d}","\frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d}",1,"-(Cos[c + d*x]/(a*d)) + Log[1 + Cos[c + d*x]]/(a*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
62,1,58,0,0.0970141,"\int \frac{\csc (c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]/(a + a*Sec[c + d*x]),x]","-\frac{\csc ^2(c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{\csc ^2(c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"-ArcTanh[Cos[c + d*x]]/(2*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - Csc[c + d*x]^2/(2*a*d)","A",6,6,19,0.3158,1,"{3872, 2706, 2606, 30, 2611, 3770}"
63,1,82,0,0.158394,"\int \frac{\csc ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^3/(a + a*Sec[c + d*x]),x]","-\frac{\csc ^4(c+d x)}{4 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}","-\frac{\csc ^4(c+d x)}{4 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}",1,"-ArcTanh[Cos[c + d*x]]/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d) - Csc[c + d*x]^4/(4*a*d)","A",7,7,21,0.3333,1,"{3872, 2835, 2606, 30, 2611, 3768, 3770}"
64,1,106,0,0.1726926,"\int \frac{\csc ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^5/(a + a*Sec[c + d*x]),x]","-\frac{\csc ^6(c+d x)}{6 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d}","-\frac{\csc ^6(c+d x)}{6 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d}",1,"-ArcTanh[Cos[c + d*x]]/(16*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(24*a*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d) - Csc[c + d*x]^6/(6*a*d)","A",8,7,21,0.3333,1,"{3872, 2835, 2606, 30, 2611, 3768, 3770}"
65,1,125,0,0.210315,"\int \frac{\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^8/(a + a*Sec[c + d*x]),x]","\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac{5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a}","\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac{5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a}",1,"(-5*x)/(128*a) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x]^3)/(48*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a*d) + Sin[c + d*x]^7/(7*a*d)","A",9,7,21,0.3333,1,"{3872, 2839, 2564, 30, 2568, 2635, 8}"
66,1,99,0,0.1774969,"\int \frac{\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}","\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}",1,"-x/(16*a) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d) + Sin[c + d*x]^5/(5*a*d)","A",8,7,21,0.3333,1,"{3872, 2839, 2564, 30, 2568, 2635, 8}"
67,1,73,0,0.15003,"\int \frac{\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a}","\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a}",1,"-x/(8*a) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) + Sin[c + d*x]^3/(3*a*d)","A",7,7,21,0.3333,1,"{3872, 2839, 2564, 30, 2568, 2635, 8}"
68,1,44,0,0.1087267,"\int \frac{\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Sin[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}","\frac{\sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}",1,"-x/(2*a) + Sin[c + d*x]/(a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",5,5,21,0.2381,1,"{3872, 2839, 2637, 2635, 8}"
69,1,37,0,0.1261,"\int \frac{\csc ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^3(c+d x)}{3 a d}","\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"Cot[c + d*x]^3/(3*a*d) - Csc[c + d*x]^3/(3*a*d)","A",6,5,21,0.2381,1,"{3872, 2839, 2606, 30, 2607}"
70,1,55,0,0.1428104,"\int \frac{\csc ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^5(c+d x)}{5 a d}","\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^5(c+d x)}{5 a d}",1,"Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) - Csc[c + d*x]^5/(5*a*d)","A",7,6,21,0.2857,1,"{3872, 2839, 2606, 30, 2607, 14}"
71,1,73,0,0.1470041,"\int \frac{\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\cot ^7(c+d x)}{7 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^7(c+d x)}{7 a d}","\frac{\cot ^7(c+d x)}{7 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^7(c+d x)}{7 a d}",1,"Cot[c + d*x]^3/(3*a*d) + (2*Cot[c + d*x]^5)/(5*a*d) + Cot[c + d*x]^7/(7*a*d) - Csc[c + d*x]^7/(7*a*d)","A",7,6,21,0.2857,1,"{3872, 2839, 2606, 30, 2607, 270}"
72,1,91,0,0.1507651,"\int \frac{\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x]),x]","\frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot ^7(c+d x)}{7 a d}+\frac{3 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^9(c+d x)}{9 a d}","\frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot ^7(c+d x)}{7 a d}+\frac{3 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^9(c+d x)}{9 a d}",1,"Cot[c + d*x]^3/(3*a*d) + (3*Cot[c + d*x]^5)/(5*a*d) + (3*Cot[c + d*x]^7)/(7*a*d) + Cot[c + d*x]^9/(9*a*d) - Csc[c + d*x]^9/(9*a*d)","A",7,6,21,0.2857,1,"{3872, 2839, 2606, 30, 2607, 270}"
73,1,109,0,0.155081,"\int \frac{\csc ^{10}(c+d x)}{a+a \sec (c+d x)} \, dx","Int[Csc[c + d*x]^10/(a + a*Sec[c + d*x]),x]","\frac{\cot ^{11}(c+d x)}{11 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^{11}(c+d x)}{11 a d}","\frac{\cot ^{11}(c+d x)}{11 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^{11}(c+d x)}{11 a d}",1,"Cot[c + d*x]^3/(3*a*d) + (4*Cot[c + d*x]^5)/(5*a*d) + (6*Cot[c + d*x]^7)/(7*a*d) + (4*Cot[c + d*x]^9)/(9*a*d) + Cot[c + d*x]^11/(11*a*d) - Csc[c + d*x]^11/(11*a*d)","A",7,6,21,0.2857,1,"{3872, 2839, 2606, 30, 2607, 270}"
74,1,137,0,0.1858019,"\int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^2,x]","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d}","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d}",1,"(4*(a - a*Cos[c + d*x])^6)/(3*a^8*d) - (4*(a - a*Cos[c + d*x])^7)/(a^9*d) + (19*(a - a*Cos[c + d*x])^8)/(4*a^10*d) - (25*(a - a*Cos[c + d*x])^9)/(9*a^11*d) + (4*(a - a*Cos[c + d*x])^10)/(5*a^12*d) - (a - a*Cos[c + d*x])^11/(11*a^13*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
75,1,114,0,0.1802021,"\int \frac{\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]","\frac{(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d}","\frac{(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d}",1,"(4*(a - a*Cos[c + d*x])^5)/(5*a^7*d) - (2*(a - a*Cos[c + d*x])^6)/(a^8*d) + (13*(a - a*Cos[c + d*x])^7)/(7*a^9*d) - (3*(a - a*Cos[c + d*x])^8)/(4*a^10*d) + (a - a*Cos[c + d*x])^9/(9*a^11*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
76,1,73,0,0.1593363,"\int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{\cos ^6(c+d x)}{3 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}","\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{\cos ^6(c+d x)}{3 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}",1,"-Cos[c + d*x]^3/(3*a^2*d) + Cos[c + d*x]^4/(2*a^2*d) - Cos[c + d*x]^6/(3*a^2*d) + Cos[c + d*x]^7/(7*a^2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 75}"
77,1,55,0,0.1541962,"\int \frac{\sin ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}",1,"-Cos[c + d*x]^3/(3*a^2*d) + Cos[c + d*x]^4/(2*a^2*d) - Cos[c + d*x]^5/(5*a^2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 43}"
78,1,66,0,0.163219,"\int \frac{\sin ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{\cos ^2(c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{2 \log (\cos (c+d x)+1)}{a^2 d}","\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{\cos ^2(c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{2 \log (\cos (c+d x)+1)}{a^2 d}",1,"(2*Cos[c + d*x])/(a^2*d) - Cos[c + d*x]^2/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - (2*Log[1 + Cos[c + d*x]])/(a^2*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 77}"
79,1,52,0,0.102093,"\int \frac{\sin (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{2 \log (\cos (c+d x)+1)}{a^2 d}","-\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{2 \log (\cos (c+d x)+1)}{a^2 d}",1,"-(Cos[c + d*x]/(a^2*d)) + 1/(d*(a^2 + a^2*Cos[c + d*x])) + (2*Log[1 + Cos[c + d*x]])/(a^2*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
80,1,60,0,0.1266293,"\int \frac{\csc (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{3}{4 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{1}{4 d (a \cos (c+d x)+a)^2}","-\frac{3}{4 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{1}{4 d (a \cos (c+d x)+a)^2}",1,"-ArcTanh[Cos[c + d*x]]/(4*a^2*d) + 1/(4*d*(a + a*Cos[c + d*x])^2) - 3/(4*d*(a^2 + a^2*Cos[c + d*x]))","A",6,5,19,0.2632,1,"{3872, 2836, 12, 88, 206}"
81,1,42,0,0.1272589,"\int \frac{\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","-\frac{2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3}","-\frac{2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3}",1,"-(a + 2*a*Cos[c + d*x])/(6*d*(1 - Cos[c + d*x])*(a + a*Cos[c + d*x])^3)","A",4,4,21,0.1905,1,"{3872, 2836, 12, 81}"
82,1,146,0,0.2169065,"\int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","\frac{a^2}{32 d (a \cos (c+d x)+a)^4}-\frac{1}{64 d \left(a^2-a^2 \cos (c+d x)\right)}-\frac{1}{32 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{a}{48 d (a \cos (c+d x)+a)^3}-\frac{1}{64 d (a-a \cos (c+d x))^2}-\frac{1}{32 d (a \cos (c+d x)+a)^2}","\frac{a^2}{32 d (a \cos (c+d x)+a)^4}-\frac{1}{64 d \left(a^2-a^2 \cos (c+d x)\right)}-\frac{1}{32 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{a}{48 d (a \cos (c+d x)+a)^3}-\frac{1}{64 d (a-a \cos (c+d x))^2}-\frac{1}{32 d (a \cos (c+d x)+a)^2}",1,"ArcTanh[Cos[c + d*x]]/(64*a^2*d) - 1/(64*d*(a - a*Cos[c + d*x])^2) + a^2/(32*d*(a + a*Cos[c + d*x])^4) - a/(48*d*(a + a*Cos[c + d*x])^3) - 1/(32*d*(a + a*Cos[c + d*x])^2) - 1/(64*d*(a^2 - a^2*Cos[c + d*x])) - 1/(32*d*(a^2 + a^2*Cos[c + d*x]))","A",6,5,21,0.2381,1,"{3872, 2836, 12, 88, 206}"
83,1,167,0,0.4400652,"\int \frac{\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sin ^7(c+d x)}{7 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac{11 x}{128 a^2}","\frac{2 \sin ^7(c+d x)}{7 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac{11 x}{128 a^2}",1,"(11*x)/(128*a^2) + (11*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) - (7*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(16*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d) - (2*Sin[c + d*x]^5)/(5*a^2*d) + (2*Sin[c + d*x]^7)/(7*a^2*d)","A",16,8,21,0.3810,1,"{3872, 2875, 2873, 2568, 2635, 8, 2564, 14}"
84,1,104,0,0.3111394,"\int \frac{\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","-\frac{\sin ^5(c+d x)}{10 a^2 d}-\frac{\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2}","-\frac{\sin ^5(c+d x)}{10 a^2 d}-\frac{\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2}",1,"(3*x)/(16*a^2) - (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(8*a^2*d) - ((a - a*Cos[c + d*x])^3*Sin[c + d*x]^3)/(6*a^5*d) - Sin[c + d*x]^5/(10*a^2*d)","A",7,6,21,0.2857,1,"{3872, 2875, 2870, 2669, 2635, 8}"
85,1,87,0,0.2325279,"\int \frac{\sin ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sin ^3(c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2}","\frac{2 \sin ^3(c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2}",1,"(7*x)/(8*a^2) - (2*Sin[c + d*x])/(a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (2*Sin[c + d*x]^3)/(3*a^2*d)","A",11,6,21,0.2857,1,"{3872, 2869, 2757, 2635, 8, 2633}"
86,1,69,0,0.3161973,"\int \frac{\sin ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sin (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{5 x}{2 a^2}","\frac{2 \sin (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{5 x}{2 a^2}",1,"(-5*x)/(2*a^2) + (2*Sin[c + d*x])/(a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (2*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x]))","A",9,8,21,0.3810,1,"{3872, 2874, 2950, 2709, 2637, 2635, 8, 2648}"
87,1,73,0,0.1995516,"\int \frac{\csc ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}",1,"-Cot[c + d*x]^3/(3*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d)","A",11,6,21,0.2857,1,"{3872, 2711, 2607, 30, 2606, 14}"
88,1,91,0,0.3449431,"\int \frac{\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}-\frac{3 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{2 \csc ^5(c+d x)}{5 a^2 d}","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}-\frac{3 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{2 \csc ^5(c+d x)}{5 a^2 d}",1,"-Cot[c + d*x]^3/(3*a^2*d) - (3*Cot[c + d*x]^5)/(5*a^2*d) - (2*Cot[c + d*x]^7)/(7*a^2*d) - (2*Csc[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d)","A",13,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 14, 2606, 270}"
89,1,109,0,0.3506795,"\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{2 \csc ^7(c+d x)}{7 a^2 d}","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{2 \csc ^7(c+d x)}{7 a^2 d}",1,"-Cot[c + d*x]^3/(3*a^2*d) - (4*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (2*Cot[c + d*x]^9)/(9*a^2*d) - (2*Csc[c + d*x]^7)/(7*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d)","A",13,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 270, 2606, 14}"
90,1,125,0,0.3672122,"\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d}","-\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d}",1,"-Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(a^2*d) - (9*Cot[c + d*x]^7)/(7*a^2*d) - (7*Cot[c + d*x]^9)/(9*a^2*d) - (2*Cot[c + d*x]^11)/(11*a^2*d) - (2*Csc[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d)","A",13,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 270, 2606, 14}"
91,1,139,0,0.1947165,"\int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d}","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d}",1,"(2*(a - a*Cos[c + d*x])^6)/(3*a^9*d) - (16*(a - a*Cos[c + d*x])^7)/(7*a^10*d) + (25*(a - a*Cos[c + d*x])^8)/(8*a^11*d) - (19*(a - a*Cos[c + d*x])^9)/(9*a^12*d) + (7*(a - a*Cos[c + d*x])^10)/(10*a^13*d) - (a - a*Cos[c + d*x])^11/(11*a^14*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
92,1,109,0,0.1786116,"\int \frac{\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^9(c+d x)}{9 a^3 d}+\frac{3 \cos ^8(c+d x)}{8 a^3 d}-\frac{2 \cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{3 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}","-\frac{\cos ^9(c+d x)}{9 a^3 d}+\frac{3 \cos ^8(c+d x)}{8 a^3 d}-\frac{2 \cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{3 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}",1,"-Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(3*a^3*d) - (2*Cos[c + d*x]^7)/(7*a^3*d) + (3*Cos[c + d*x]^8)/(8*a^3*d) - Cos[c + d*x]^9/(9*a^3*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 75}"
93,1,73,0,0.1648185,"\int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}","\frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}",1,"-Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(2*a^3*d) + Cos[c + d*x]^7/(7*a^3*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 43}"
94,1,102,0,0.1817046,"\int \frac{\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^5(c+d x)}{5 a^3 d}+\frac{3 \cos ^4(c+d x)}{4 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{2 \cos ^2(c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{4 \log (\cos (c+d x)+1)}{a^3 d}","-\frac{\cos ^5(c+d x)}{5 a^3 d}+\frac{3 \cos ^4(c+d x)}{4 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{2 \cos ^2(c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{4 \log (\cos (c+d x)+1)}{a^3 d}",1,"(-4*Cos[c + d*x])/(a^3*d) + (2*Cos[c + d*x]^2)/(a^3*d) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (3*Cos[c + d*x]^4)/(4*a^3*d) - Cos[c + d*x]^5/(5*a^3*d) + (4*Log[1 + Cos[c + d*x]])/(a^3*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 88}"
95,1,89,0,0.1837355,"\int \frac{\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^2(c+d x)}{2 a^3 d}+\frac{5 \cos (c+d x)}{a^3 d}-\frac{2}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \log (\cos (c+d x)+1)}{a^3 d}","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^2(c+d x)}{2 a^3 d}+\frac{5 \cos (c+d x)}{a^3 d}-\frac{2}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \log (\cos (c+d x)+1)}{a^3 d}",1,"(5*Cos[c + d*x])/(a^3*d) - (3*Cos[c + d*x]^2)/(2*a^3*d) + Cos[c + d*x]^3/(3*a^3*d) - 2/(d*(a^3 + a^3*Cos[c + d*x])) - (7*Log[1 + Cos[c + d*x]])/(a^3*d)","A",5,4,21,0.1905,1,"{3872, 2836, 12, 77}"
96,1,75,0,0.1167474,"\int \frac{\sin (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos (c+d x)}{a^3 d}+\frac{3}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \log (\cos (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \cos (c+d x)+a)^2}","-\frac{\cos (c+d x)}{a^3 d}+\frac{3}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \log (\cos (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \cos (c+d x)+a)^2}",1,"-(Cos[c + d*x]/(a^3*d)) - 1/(2*a*d*(a + a*Cos[c + d*x])^2) + 3/(d*(a^3 + a^3*Cos[c + d*x])) + (3*Log[1 + Cos[c + d*x]])/(a^3*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
97,1,82,0,0.1511339,"\int \frac{\csc (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]/(a + a*Sec[c + d*x])^3,x]","-\frac{7}{8 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{5}{8 a d (a \cos (c+d x)+a)^2}-\frac{1}{6 d (a \cos (c+d x)+a)^3}","-\frac{7}{8 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{5}{8 a d (a \cos (c+d x)+a)^2}-\frac{1}{6 d (a \cos (c+d x)+a)^3}",1,"-ArcTanh[Cos[c + d*x]]/(8*a^3*d) - 1/(6*d*(a + a*Cos[c + d*x])^3) + 5/(8*a*d*(a + a*Cos[c + d*x])^2) - 7/(8*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,19,0.2632,1,"{3872, 2836, 12, 88, 206}"
98,1,126,0,0.134334,"\int \frac{\csc ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","-\frac{1}{32 d \left(a^3-a^3 \cos (c+d x)\right)}-\frac{1}{16 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \cos (c+d x)+a)^4}+\frac{1}{6 d (a \cos (c+d x)+a)^3}-\frac{3}{32 a d (a \cos (c+d x)+a)^2}","-\frac{1}{32 d \left(a^3-a^3 \cos (c+d x)\right)}-\frac{1}{16 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \cos (c+d x)+a)^4}+\frac{1}{6 d (a \cos (c+d x)+a)^3}-\frac{3}{32 a d (a \cos (c+d x)+a)^2}",1,"ArcTanh[Cos[c + d*x]]/(32*a^3*d) - a/(16*d*(a + a*Cos[c + d*x])^4) + 1/(6*d*(a + a*Cos[c + d*x])^3) - 3/(32*a*d*(a + a*Cos[c + d*x])^2) - 1/(32*d*(a^3 - a^3*Cos[c + d*x])) - 1/(16*d*(a^3 + a^3*Cos[c + d*x]))","A",5,4,21,0.1905,1,"{3872, 2707, 88, 206}"
99,1,128,0,0.2056671,"\int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","-\frac{a^2}{40 d (a \cos (c+d x)+a)^5}-\frac{3}{128 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}+\frac{3 a}{64 d (a \cos (c+d x)+a)^4}-\frac{1}{128 a d (a-a \cos (c+d x))^2}-\frac{1}{64 a d (a \cos (c+d x)+a)^2}","-\frac{a^2}{40 d (a \cos (c+d x)+a)^5}-\frac{3}{128 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}+\frac{3 a}{64 d (a \cos (c+d x)+a)^4}-\frac{1}{128 a d (a-a \cos (c+d x))^2}-\frac{1}{64 a d (a \cos (c+d x)+a)^2}",1,"(3*ArcTanh[Cos[c + d*x]])/(128*a^3*d) - 1/(128*a*d*(a - a*Cos[c + d*x])^2) - a^2/(40*d*(a + a*Cos[c + d*x])^5) + (3*a)/(64*d*(a + a*Cos[c + d*x])^4) - 1/(64*a*d*(a + a*Cos[c + d*x])^2) - 3/(128*d*(a^3 + a^3*Cos[c + d*x]))","A",6,5,21,0.2381,1,"{3872, 2836, 12, 88, 206}"
100,1,157,0,0.460523,"\int \frac{\sin ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","\frac{3 \sin ^7(c+d x)}{7 a^3 d}-\frac{7 \sin ^5(c+d x)}{5 a^3 d}+\frac{4 \sin ^3(c+d x)}{3 a^3 d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a^3 d}+\frac{23 \sin (c+d x) \cos ^5(c+d x)}{48 a^3 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{192 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3}","\frac{3 \sin ^7(c+d x)}{7 a^3 d}-\frac{7 \sin ^5(c+d x)}{5 a^3 d}+\frac{4 \sin ^3(c+d x)}{3 a^3 d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a^3 d}+\frac{23 \sin (c+d x) \cos ^5(c+d x)}{48 a^3 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{192 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3}",1,"(-29*x)/(128*a^3) - (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(192*a^3*d) + (23*Cos[c + d*x]^5*Sin[c + d*x])/(48*a^3*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a^3*d) + (4*Sin[c + d*x]^3)/(3*a^3*d) - (7*Sin[c + d*x]^5)/(5*a^3*d) + (3*Sin[c + d*x]^7)/(7*a^3*d)","A",19,9,21,0.4286,1,"{3872, 2875, 2873, 2564, 14, 2568, 2635, 8, 270}"
101,1,129,0,0.2913313,"\int \frac{\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{3 \sin ^5(c+d x)}{5 a^3 d}-\frac{7 \sin ^3(c+d x)}{3 a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac{23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3}","\frac{3 \sin ^5(c+d x)}{5 a^3 d}-\frac{7 \sin ^3(c+d x)}{3 a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac{23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3}",1,"(-23*x)/(16*a^3) + (4*Sin[c + d*x])/(a^3*d) - (23*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (23*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a^3*d) - (7*Sin[c + d*x]^3)/(3*a^3*d) + (3*Sin[c + d*x]^5)/(5*a^3*d)","A",15,6,21,0.2857,1,"{3872, 2869, 2757, 2633, 2635, 8}"
102,1,108,0,0.3184773,"\int \frac{\sin ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{\sin ^3(c+d x)}{a^3 d}-\frac{7 \sin (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}+\frac{19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{4 \sin (c+d x)}{a^3 d (\cos (c+d x)+1)}+\frac{51 x}{8 a^3}","\frac{\sin ^3(c+d x)}{a^3 d}-\frac{7 \sin (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}+\frac{19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{4 \sin (c+d x)}{a^3 d (\cos (c+d x)+1)}+\frac{51 x}{8 a^3}",1,"(51*x)/(8*a^3) - (7*Sin[c + d*x])/(a^3*d) + (19*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*d) - (4*Sin[c + d*x])/(a^3*d*(1 + Cos[c + d*x])) + Sin[c + d*x]^3/(a^3*d)","A",13,8,21,0.3810,1,"{3872, 2875, 2872, 2648, 2637, 2635, 8, 2633}"
103,1,97,0,0.3097808,"\int \frac{\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{3 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac{2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac{11 x}{2 a^3}","\frac{3 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac{2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac{11 x}{2 a^3}",1,"(-11*x)/(2*a^3) + (3*Sin[c + d*x])/(a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (2*Sin[c + d*x])/(3*a^3*d*(1 + Cos[c + d*x])^2) + (19*Sin[c + d*x])/(3*a^3*d*(1 + Cos[c + d*x]))","A",10,8,21,0.3810,1,"{3872, 2874, 2966, 2637, 2635, 8, 2650, 2648}"
104,1,89,0,0.3669355,"\int \frac{\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{7 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^3(c+d x)}{a^3 d}","\frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{7 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^3(c+d x)}{a^3 d}",1,"(3*Cot[c + d*x]^5)/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^3/(a^3*d) + (7*Csc[c + d*x]^5)/(5*a^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)","A",15,8,21,0.3810,1,"{3872, 2875, 2873, 2607, 30, 2606, 270, 14}"
105,1,103,0,0.3787056,"\int \frac{\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{\csc ^7(c+d x)}{a^3 d}-\frac{3 \csc ^5(c+d x)}{5 a^3 d}","\frac{4 \cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{\csc ^7(c+d x)}{a^3 d}-\frac{3 \csc ^5(c+d x)}{5 a^3 d}",1,"(3*Cot[c + d*x]^5)/(5*a^3*d) + Cot[c + d*x]^7/(a^3*d) + (4*Cot[c + d*x]^9)/(9*a^3*d) - (3*Csc[c + d*x]^5)/(5*a^3*d) + Csc[c + d*x]^7/(a^3*d) - (4*Csc[c + d*x]^9)/(9*a^3*d)","A",16,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 14, 2606, 270}"
106,1,127,0,0.4080487,"\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{11 \cot ^9(c+d x)}{9 a^3 d}+\frac{10 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{7 \csc ^9(c+d x)}{9 a^3 d}-\frac{3 \csc ^7(c+d x)}{7 a^3 d}","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{11 \cot ^9(c+d x)}{9 a^3 d}+\frac{10 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{7 \csc ^9(c+d x)}{9 a^3 d}-\frac{3 \csc ^7(c+d x)}{7 a^3 d}",1,"(3*Cot[c + d*x]^5)/(5*a^3*d) + (10*Cot[c + d*x]^7)/(7*a^3*d) + (11*Cot[c + d*x]^9)/(9*a^3*d) + (4*Cot[c + d*x]^11)/(11*a^3*d) - (3*Csc[c + d*x]^7)/(7*a^3*d) + (7*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*d)","A",16,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 270, 2606, 14}"
107,1,145,0,0.4193211,"\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","\frac{4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d}","\frac{4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d}",1,"(3*Cot[c + d*x]^5)/(5*a^3*d) + (13*Cot[c + d*x]^7)/(7*a^3*d) + (7*Cot[c + d*x]^9)/(3*a^3*d) + (15*Cot[c + d*x]^11)/(11*a^3*d) + (4*Cot[c + d*x]^13)/(13*a^3*d) - Csc[c + d*x]^9/(3*a^3*d) + (7*Csc[c + d*x]^11)/(11*a^3*d) - (4*Csc[c + d*x]^13)/(13*a^3*d)","A",16,7,21,0.3333,1,"{3872, 2875, 2873, 2607, 270, 2606, 14}"
108,1,157,0,0.2013283,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^{5/2} \, dx","Int[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2),x]","-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{6 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{2 a e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}","-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{6 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{2 a e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}",1,"-((a*e^(5/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (a*e^(5/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (6*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*e*(e*Sin[c + d*x])^(3/2))/(3*d) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d)","A",11,11,23,0.4783,1,"{3872, 2838, 2564, 321, 329, 298, 203, 206, 2635, 2640, 2639}"
109,1,154,0,0.1998854,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx","Int[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2),x]","\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}","\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}",1,"(a*e^(3/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (a*e^(3/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*a*e*Sqrt[e*Sin[c + d*x]])/d - (2*a*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d)","A",11,11,23,0.4783,1,"{3872, 2838, 2564, 321, 329, 212, 206, 203, 2635, 2642, 2641}"
110,1,104,0,0.1502441,"\int (a+a \sec (c+d x)) \sqrt{e \sin (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]],x]","-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}","-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}",1,"-((a*Sqrt[e]*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (a*Sqrt[e]*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])","A",9,9,23,0.3913,1,"{3872, 2838, 2564, 329, 298, 203, 206, 2640, 2639}"
111,1,103,0,0.151885,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])/Sqrt[e*Sin[c + d*x]],x]","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}",1,"(a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]])","A",9,9,23,0.3913,1,"{3872, 2838, 2564, 329, 212, 206, 203, 2642, 2641}"
112,1,155,0,0.199393,"\int \frac{a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(3/2),x]","-\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}","-\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}",1,"-((a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2))) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2)) - (2*a)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]])","A",11,11,23,0.4783,1,"{3872, 2838, 2564, 325, 329, 298, 203, 206, 2636, 2640, 2639}"
113,1,160,0,0.2005852,"\int \frac{a+a \sec (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(5/2),x]","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}",1,"(a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) - (2*a)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]])","A",11,11,23,0.4783,1,"{3872, 2838, 2564, 325, 329, 212, 206, 203, 2636, 2642, 2641}"
114,1,194,0,0.3824372,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2),x]","-\frac{2 a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{9 a^2 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}","-\frac{2 a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{9 a^2 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}",1,"(-2*a^2*e^(5/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a^2*e^(5/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d - (9*a^2*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (4*a^2*e*(e*Sin[c + d*x])^(3/2))/(3*d) - (2*a^2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d) + (a^2*e*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/d","A",15,12,25,0.4800,1,"{3872, 2873, 2635, 2640, 2639, 2564, 321, 329, 298, 203, 206, 2566}"
115,1,192,0,0.3800842,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2),x]","\frac{2 a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{4 a^2 e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{a^2 e \sec (c+d x) \sqrt{e \sin (c+d x)}}{d}","\frac{2 a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{4 a^2 e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{a^2 e \sec (c+d x) \sqrt{e \sin (c+d x)}}{d}",1,"(2*a^2*e^(3/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a^2*e^(3/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d - (a^2*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (4*a^2*e*Sqrt[e*Sin[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d) + (a^2*e*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/d","A",15,12,25,0.4800,1,"{3872, 2873, 2635, 2642, 2641, 2564, 321, 329, 212, 206, 203, 2566}"
116,1,138,0,0.3073678,"\int (a+a \sec (c+d x))^2 \sqrt{e \sin (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]],x]","-\frac{2 a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}","-\frac{2 a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}",1,"(-2*a^2*Sqrt[e]*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a^2*Sqrt[e]*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (a^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (a^2*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/(d*e)","A",13,10,25,0.4000,1,"{3872, 2873, 2640, 2639, 2564, 329, 298, 203, 206, 2571}"
117,1,139,0,0.3072616,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^2/Sqrt[e*Sin[c + d*x]],x]","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}",1,"(2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (3*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (a^2*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/(d*e)","A",13,10,25,0.4000,1,"{3872, 2873, 2642, 2641, 2564, 329, 212, 206, 203, 2571}"
118,1,224,0,0.4240207,"\int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}","-\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}",1,"(-2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2)) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2)) - (4*a^2)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a^2*Sec[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) + (3*a^2*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/(d*e^3)","A",16,13,25,0.5200,1,"{3872, 2873, 2636, 2640, 2639, 2564, 325, 329, 298, 203, 206, 2570, 2571}"
119,1,234,0,0.4191214,"\int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{5 a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{7 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{5 a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{7 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}",1,"(2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) - (4*a^2)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2*Sec[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (7*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) + (5*a^2*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)","A",16,13,25,0.5200,1,"{3872, 2873, 2636, 2642, 2641, 2564, 325, 329, 212, 206, 203, 2570, 2571}"
120,1,139,0,0.2796314,"\int \frac{(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x]),x]","\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a d}-\frac{2 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a d}-\frac{4 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2}}{5 a d}","\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a d}-\frac{2 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a d}-\frac{4 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2}}{5 a d}",1,"(-4*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a*d*Sqrt[e*Sin[c + d*x]]) - (2*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a*d) + (2*e^3*Cos[c + d*x]^3*Sqrt[e*Sin[c + d*x]])/(7*a*d) + (2*e*(e*Sin[c + d*x])^(5/2))/(5*a*d)","A",8,8,25,0.3200,1,"{3872, 2839, 2564, 30, 2568, 2569, 2642, 2641}"
121,1,104,0,0.2199417,"\int \frac{(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","-\frac{4 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d \sqrt{\sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d}","-\frac{4 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d \sqrt{\sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d}",1,"(-4*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a*d*Sqrt[Sin[c + d*x]]) + (2*e*(e*Sin[c + d*x])^(3/2))/(3*a*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a*d)","A",7,7,25,0.2800,1,"{3872, 2839, 2564, 30, 2569, 2640, 2639}"
122,1,102,0,0.2224651,"\int \frac{(e \sin (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}+\frac{2 e \sqrt{e \sin (c+d x)}}{a d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a d}","-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}+\frac{2 e \sqrt{e \sin (c+d x)}}{a d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a d}",1,"(-4*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a*d*Sqrt[e*Sin[c + d*x]]) + (2*e*Sqrt[e*Sin[c + d*x]])/(a*d) - (2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a*d)","A",7,7,25,0.2800,1,"{3872, 2839, 2564, 30, 2569, 2642, 2641}"
123,1,95,0,0.2077126,"\int \frac{\sqrt{e \sin (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}","-\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}",1,"(-2*e)/(a*d*Sqrt[e*Sin[c + d*x]]) + (2*e*Cos[c + d*x])/(a*d*Sqrt[e*Sin[c + d*x]]) + (4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]])","A",7,7,25,0.2800,1,"{3872, 2839, 2564, 30, 2567, 2640, 2639}"
124,1,101,0,0.2113532,"\int \frac{1}{(a+a \sec (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","-\frac{2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac{2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}","-\frac{2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac{2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}",1,"(-2*e)/(3*a*d*(e*Sin[c + d*x])^(3/2)) + (2*e*Cos[c + d*x])/(3*a*d*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a*d*Sqrt[e*Sin[c + d*x]])","A",7,7,25,0.2800,1,"{3872, 2839, 2564, 30, 2567, 2642, 2641}"
125,1,135,0,0.2483371,"\int \frac{1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]","-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d e^2 \sqrt{\sin (c+d x)}}-\frac{2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac{2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{5 a d e \sqrt{e \sin (c+d x)}}","-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d e^2 \sqrt{\sin (c+d x)}}-\frac{2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac{2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{5 a d e \sqrt{e \sin (c+d x)}}",1,"(-2*e)/(5*a*d*(e*Sin[c + d*x])^(5/2)) + (2*e*Cos[c + d*x])/(5*a*d*(e*Sin[c + d*x])^(5/2)) - (4*Cos[c + d*x])/(5*a*d*e*Sqrt[e*Sin[c + d*x]]) - (4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a*d*e^2*Sqrt[Sin[c + d*x]])","A",8,8,25,0.3200,1,"{3872, 2839, 2564, 30, 2567, 2636, 2640, 2639}"
126,1,135,0,0.2503527,"\int \frac{1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}","\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}",1,"(-2*e)/(7*a*d*(e*Sin[c + d*x])^(7/2)) + (2*e*Cos[c + d*x])/(7*a*d*(e*Sin[c + d*x])^(7/2)) - (4*Cos[c + d*x])/(21*a*d*e*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a*d*e^2*Sqrt[e*Sin[c + d*x]])","A",8,8,25,0.3200,1,"{3872, 2839, 2564, 30, 2567, 2636, 2642, 2641}"
127,1,162,0,0.5510932,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{4 e^3 \sqrt{e \sin (c+d x)}}{a^2 d}+\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a^2 d}+\frac{26 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}+\frac{52 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{5/2}}{5 a^2 d}","-\frac{4 e^3 \sqrt{e \sin (c+d x)}}{a^2 d}+\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a^2 d}+\frac{26 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}+\frac{52 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{5/2}}{5 a^2 d}",1,"(52*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]]) - (4*e^3*Sqrt[e*Sin[c + d*x]])/(a^2*d) + (26*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) + (2*e^3*Cos[c + d*x]^3*Sqrt[e*Sin[c + d*x]])/(7*a^2*d) + (4*e*(e*Sin[c + d*x])^(5/2))/(5*a^2*d)","A",14,8,25,0.3200,1,"{3872, 2875, 2873, 2569, 2642, 2641, 2564, 14}"
128,1,187,0,0.5961517,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{4 e^3}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos (c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{44 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac{12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}","\frac{4 e^3}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos (c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{44 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac{12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}",1,"(4*e^3)/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (2*e^3*Cos[c + d*x])/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (2*e^3*Cos[c + d*x]^3)/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (44*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]]) + (4*e*(e*Sin[c + d*x])^(3/2))/(3*a^2*d) - (12*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a^2*d)","A",14,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2640, 2639, 2564, 14, 2569}"
129,1,189,0,0.5935636,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e \sqrt{e \sin (c+d x)}}{a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}","\frac{4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e \sqrt{e \sin (c+d x)}}{a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}",1,"(4*e^3)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (2*e^3*Cos[c + d*x])/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (2*e^3*Cos[c + d*x]^3)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (4*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (4*e*Sqrt[e*Sin[c + d*x]])/(a^2*d) - (4*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a^2*d)","A",14,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2642, 2641, 2564, 14, 2569}"
130,1,188,0,0.5896787,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\frac{4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 e}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{16 e \cos (c+d x)}{5 a^2 d \sqrt{e \sin (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}","\frac{4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 e}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{16 e \cos (c+d x)}{5 a^2 d \sqrt{e \sin (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}",1,"(4*e^3)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (2*e^3*Cos[c + d*x])/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (2*e^3*Cos[c + d*x]^3)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (4*e)/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (16*e*Cos[c + d*x])/(5*a^2*d*Sqrt[e*Sin[c + d*x]]) + (28*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]])","A",15,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2636, 2640, 2639, 2564, 14}"
131,1,190,0,0.5908832,"\int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),x]","\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}","\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}",1,"(4*e^3)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (2*e^3*Cos[c + d*x])/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (2*e^3*Cos[c + d*x]^3)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (4*e)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) + (16*e*Cos[c + d*x])/(21*a^2*d*(e*Sin[c + d*x])^(3/2)) + (20*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]])","A",15,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2636, 2642, 2641, 2564, 14}"
132,1,224,0,0.6638001,"\int \frac{1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),x]","\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}","\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}",1,"(4*e^3)/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (2*e^3*Cos[c + d*x])/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (2*e^3*Cos[c + d*x]^3)/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (4*e)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) + (16*e*Cos[c + d*x])/(45*a^2*d*(e*Sin[c + d*x])^(5/2)) - (4*Cos[c + d*x])/(15*a^2*d*e*Sqrt[e*Sin[c + d*x]]) - (4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(15*a^2*d*e^2*Sqrt[Sin[c + d*x]])","A",17,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2636, 2640, 2639, 2564, 14}"
133,1,224,0,0.6701052,"\int \frac{1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","\frac{4 e^3}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos ^3(c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos (c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{231 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 e}{7 a^2 d (e \sin (c+d x))^{7/2}}+\frac{16 e \cos (c+d x)}{77 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{231 a^2 d e (e \sin (c+d x))^{3/2}}","\frac{4 e^3}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos ^3(c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos (c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{231 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 e}{7 a^2 d (e \sin (c+d x))^{7/2}}+\frac{16 e \cos (c+d x)}{77 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{231 a^2 d e (e \sin (c+d x))^{3/2}}",1,"(4*e^3)/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (2*e^3*Cos[c + d*x])/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (2*e^3*Cos[c + d*x]^3)/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (4*e)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) + (16*e*Cos[c + d*x])/(77*a^2*d*(e*Sin[c + d*x])^(7/2)) - (4*Cos[c + d*x])/(231*a^2*d*e*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*a^2*d*e^2*Sqrt[e*Sin[c + d*x]])","A",17,9,25,0.3600,1,"{3872, 2875, 2873, 2567, 2636, 2642, 2641, 2564, 14}"
134,1,247,0,0.3526408,"\int (a+a \sec (c+d x))^3 (e \sin (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^3*(e*Sin[c + d*x])^m,x]","\frac{3 a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}","\frac{3 a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (3*a^3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))","A",9,6,23,0.2609,1,"{3872, 2873, 2643, 2564, 364, 2577}"
135,1,195,0,0.2851484,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^m,x]","\frac{2 a^2 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}","\frac{2 a^2 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))","A",7,6,23,0.2609,1,"{3872, 2873, 2643, 2564, 364, 2577}"
136,1,119,0,0.1430236,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]","\frac{a (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}","\frac{a (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))","A",5,5,21,0.2381,1,"{3872, 2838, 2564, 364, 2643}"
137,1,100,0,0.1988089,"\int \frac{(e \sin (c+d x))^m}{a+a \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x]),x]","\frac{e \cos (c+d x) (e \sin (c+d x))^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m) \sqrt{\cos ^2(c+d x)}}-\frac{e (e \sin (c+d x))^{m-1}}{a d (1-m)}","\frac{e \cos (c+d x) (e \sin (c+d x))^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m) \sqrt{\cos ^2(c+d x)}}-\frac{e (e \sin (c+d x))^{m-1}}{a d (1-m)}",1,"-((e*(e*Sin[c + d*x])^(-1 + m))/(a*d*(1 - m))) + (e*Cos[c + d*x]*Hypergeometric2F1[-1/2, (-1 + m)/2, (1 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-1 + m))/(a*d*(1 - m)*Sqrt[Cos[c + d*x]^2])","A",5,5,23,0.2174,1,"{3872, 2839, 2564, 30, 2577}"
138,1,207,0,0.5257465,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^2,x]","-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}+\frac{2 e^3 (e \sin (c+d x))^{m-3}}{a^2 d (3-m)}-\frac{2 e (e \sin (c+d x))^{m-1}}{a^2 d (1-m)}","-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}+\frac{2 e^3 (e \sin (c+d x))^{m-3}}{a^2 d (3-m)}-\frac{2 e (e \sin (c+d x))^{m-1}}{a^2 d (1-m)}",1,"(2*e^3*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)) - (e^3*Cos[c + d*x]*Hypergeometric2F1[-3/2, (-3 + m)/2, (-1 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)*Sqrt[Cos[c + d*x]^2]) - (e^3*Cos[c + d*x]*Hypergeometric2F1[-1/2, (-3 + m)/2, (-1 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)*Sqrt[Cos[c + d*x]^2]) - (2*e*(e*Sin[c + d*x])^(-1 + m))/(a^2*d*(1 - m))","A",9,6,23,0.2609,1,"{3872, 2875, 2873, 2577, 2564, 14}"
139,1,236,0,0.6353269,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^3,x]","\frac{e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{5}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}+\frac{3 e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{3}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}-\frac{4 e^5 (e \sin (c+d x))^{m-5}}{a^3 d (5-m)}+\frac{7 e^3 (e \sin (c+d x))^{m-3}}{a^3 d (3-m)}-\frac{3 e (e \sin (c+d x))^{m-1}}{a^3 d (1-m)}","\frac{e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{5}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}+\frac{3 e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{3}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}-\frac{4 e^5 (e \sin (c+d x))^{m-5}}{a^3 d (5-m)}+\frac{7 e^3 (e \sin (c+d x))^{m-3}}{a^3 d (3-m)}-\frac{3 e (e \sin (c+d x))^{m-1}}{a^3 d (1-m)}",1,"(-4*e^5*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) + (e^5*Cos[c + d*x]*Hypergeometric2F1[-5/2, (-5 + m)/2, (-3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m)*Sqrt[Cos[c + d*x]^2]) + (3*e^5*Cos[c + d*x]*Hypergeometric2F1[-3/2, (-5 + m)/2, (-3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m)*Sqrt[Cos[c + d*x]^2]) + (7*e^3*(e*Sin[c + d*x])^(-3 + m))/(a^3*d*(3 - m)) - (3*e*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(1 - m))","A",12,7,23,0.3043,1,"{3872, 2875, 2873, 2564, 14, 2577, 270}"
140,1,106,0,0.37476,"\int (a+a \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m,x]","\frac{2 a e \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(-\frac{1}{2};\frac{1-m}{2},\frac{1}{2} (-m-2);\frac{1}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}","\frac{2 a e \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(-\frac{1}{2};\frac{1-m}{2},\frac{1}{2} (-m-2);\frac{1}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}",1,"(2*a*e*AppellF1[-1/2, (1 - m)/2, (-2 - m)/2, 1/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^(-1 + m))/(d*(1 + Cos[c + d*x])^(m/2))","A",5,4,25,0.1600,1,"{3876, 2886, 135, 133}"
141,1,107,0,0.313668,"\int \sqrt{a+a \sec (c+d x)} (e \sin (c+d x))^m \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^m,x]","-\frac{2 e \cos (c+d x) \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(\frac{1}{2};\frac{1-m}{2},-\frac{m}{2};\frac{3}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}","-\frac{2 e \cos (c+d x) \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(\frac{1}{2};\frac{1-m}{2},-\frac{m}{2};\frac{3}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}",1,"(-2*e*AppellF1[1/2, (1 - m)/2, -m/2, 3/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^(-1 + m))/(d*(1 + Cos[c + d*x])^(m/2))","A",5,4,25,0.1600,1,"{3876, 2886, 135, 133}"
142,1,115,0,0.3249604,"\int \frac{(e \sin (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(e*Sin[c + d*x])^m/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{3}{2};\frac{1-m}{2},\frac{2-m}{2};\frac{5}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{3 d \sqrt{a \sec (c+d x)+a}}","-\frac{2 e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{3}{2};\frac{1-m}{2},\frac{2-m}{2};\frac{5}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(-2*e*AppellF1[3/2, (1 - m)/2, (2 - m)/2, 5/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*(1 + Cos[c + d*x])^(1 - m/2)*(e*Sin[c + d*x])^(-1 + m))/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,25,0.1600,1,"{3876, 2886, 135, 133}"
143,1,120,0,0.3735472,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{2 e \cos ^2(c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{5}{2};\frac{1-m}{2},\frac{4-m}{2};\frac{7}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{5 a d \sqrt{a \sec (c+d x)+a}}","-\frac{2 e \cos ^2(c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{5}{2};\frac{1-m}{2},\frac{4-m}{2};\frac{7}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{5 a d \sqrt{a \sec (c+d x)+a}}",1,"(-2*e*AppellF1[5/2, (1 - m)/2, (4 - m)/2, 7/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]^2*(1 + Cos[c + d*x])^(1 - m/2)*(e*Sin[c + d*x])^(-1 + m))/(5*a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,4,25,0.1600,1,"{3876, 2886, 135, 133}"
144,1,130,0,0.276536,"\int (a+a \sec (c+d x))^n (e \sin (c+d x))^m \, dx","Int[(a + a*Sec[c + d*x])^n*(e*Sin[c + d*x])^m,x]","-\frac{e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac{1}{2} (-m-2 n+1)} F_1\left(1-n;\frac{1-m}{2},\frac{1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}","-\frac{e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac{1}{2} (-m-2 n+1)} F_1\left(1-n;\frac{1-m}{2},\frac{1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}",1,"-((e*AppellF1[1 - n, (1 - m)/2, (1 - m - 2*n)/2, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*(1 + Cos[c + d*x])^((1 - m - 2*n)/2)*(a + a*Sec[c + d*x])^n*(e*Sin[c + d*x])^(-1 + m))/(d*(1 - n)))","A",5,4,23,0.1739,1,"{3876, 2886, 135, 133}"
145,1,180,0,0.1687003,"\int (a+a \sec (c+d x))^n \sin ^7(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^7,x]","-\frac{(3-n) (8-n) (16-n) (a \sec (c+d x)+a)^{n+4} \, _2F_1(6,n+4;n+5;\sec (c+d x)+1)}{42 a^4 d (1-n) (n+4)}+\frac{\cos ^7(c+d x) \left(6 (8-n)-\left(n^2-25 n+108\right) \sec (c+d x)\right) (a \sec (c+d x)+a)^{n+4}}{42 a^4 d (1-n)}-\frac{\cos ^7(c+d x) (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{n+4}}{a^4 d (1-n)}","-\frac{(3-n) (8-n) (16-n) (a \sec (c+d x)+a)^{n+4} \, _2F_1(6,n+4;n+5;\sec (c+d x)+1)}{42 a^4 d (1-n) (n+4)}+\frac{\cos ^7(c+d x) \left(6 (8-n)-\left(n^2-25 n+108\right) \sec (c+d x)\right) (a \sec (c+d x)+a)^{n+4}}{42 a^4 d (1-n)}-\frac{\cos ^7(c+d x) (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{n+4}}{a^4 d (1-n)}",1,"-((3 - n)*(8 - n)*(16 - n)*Hypergeometric2F1[6, 4 + n, 5 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(4 + n))/(42*a^4*d*(1 - n)*(4 + n)) - (Cos[c + d*x]^7*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(1 - n)) + (Cos[c + d*x]^7*(a + a*Sec[c + d*x])^(4 + n)*(6*(8 - n) - (108 - 25*n + n^2)*Sec[c + d*x]))/(42*a^4*d*(1 - n))","A",4,4,21,0.1905,1,"{3873, 100, 145, 65}"
146,1,123,0,0.1082522,"\int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^5,x]","\frac{\left(n^2-13 n+32\right) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac{\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac{(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d}","\frac{\left(n^2-13 n+32\right) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac{\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac{(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d}",1,"((12 - n)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3 + n))/(20*a^3*d) - (Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3 + n))/(5*a^3*d) + ((32 - 13*n + n^2)*Hypergeometric2F1[4, 3 + n, 4 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3 + n))/(20*a^3*d*(3 + n))","A",4,4,21,0.1905,1,"{3873, 89, 78, 65}"
147,1,83,0,0.0726852,"\int (a+a \sec (c+d x))^n \sin ^3(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^3,x]","\frac{\cos ^3(c+d x) (a \sec (c+d x)+a)^{n+2}}{3 a^2 d}-\frac{(4-n) (a \sec (c+d x)+a)^{n+2} \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)}{3 a^2 d (n+2)}","\frac{\cos ^3(c+d x) (a \sec (c+d x)+a)^{n+2}}{3 a^2 d}-\frac{(4-n) (a \sec (c+d x)+a)^{n+2} \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)}{3 a^2 d (n+2)}",1,"(Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(2 + n))/(3*a^2*d) - ((4 - n)*Hypergeometric2F1[3, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2 + n))/(3*a^2*d*(2 + n))","A",3,3,21,0.1429,1,"{3873, 78, 65}"
148,1,42,0,0.0373294,"\int (a+a \sec (c+d x))^n \sin (c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x],x]","\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(2,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}","\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(2,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}",1,"(Hypergeometric2F1[2, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1 + n))/(a*d*(1 + n))","A",2,2,19,0.1053,1,"{3873, 65}"
149,1,40,0,0.0459735,"\int \csc (c+d x) (a+a \sec (c+d x))^n \, dx","Int[Csc[c + d*x]*(a + a*Sec[c + d*x])^n,x]","-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}","-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}",1,"-(Hypergeometric2F1[1, n, 1 + n, (1 + Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^n)/(2*d*n)","A",2,2,19,0.1053,1,"{3873, 68}"
150,1,112,0,0.0968238,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^3*(a + a*Sec[c + d*x])^n,x]","-\frac{(n+2) (a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{8 d n}-\frac{a (2-n) (a \sec (c+d x)+a)^{n-1}}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}","-\frac{(n+2) (a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{8 d n}-\frac{a (2-n) (a \sec (c+d x)+a)^{n-1}}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}",1,"-(a*(2 - n)*(a + a*Sec[c + d*x])^(-1 + n))/(4*d*(1 - n)) + (a*(a + a*Sec[c + d*x])^(-1 + n))/(2*d*(1 - Sec[c + d*x])) - ((2 + n)*Hypergeometric2F1[1, n, 1 + n, (1 + Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^n)/(8*d*n)","A",4,4,21,0.1905,1,"{3873, 89, 79, 68}"
151,1,240,0,0.2239673,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^n,x]","\frac{a^2 \left(n^2+9 n+12\right) (a \sec (c+d x)+a)^{n-2} \, _2F_1\left(1,n-2;n-1;\frac{1}{2} (\sec (c+d x)+1)\right)}{16 d (2-n)}-\frac{a^2 \left(-2 (1-n) (n+6) \sec (c+d x)-n^3-7 n^2+4 n+12\right) (a \sec (c+d x)+a)^{n-2}}{8 d \left(n^2-3 n+2\right) (1-\sec (c+d x))}-\frac{a^2 \sec ^3(c+d x) (a \sec (c+d x)+a)^{n-2}}{d (1-n) (1-\sec (c+d x))^2}+\frac{a^2 (n+3) \sec ^2(c+d x) (a \sec (c+d x)+a)^{n-2}}{4 d (1-n) (1-\sec (c+d x))^2}","\frac{a^2 \left(n^2+9 n+12\right) (a \sec (c+d x)+a)^{n-2} \, _2F_1\left(1,n-2;n-1;\frac{1}{2} (\sec (c+d x)+1)\right)}{16 d (2-n)}-\frac{a^2 \left(-2 (1-n) (n+6) \sec (c+d x)-n^3-7 n^2+4 n+12\right) (a \sec (c+d x)+a)^{n-2}}{8 d \left(n^2-3 n+2\right) (1-\sec (c+d x))}-\frac{a^2 \sec ^3(c+d x) (a \sec (c+d x)+a)^{n-2}}{d (1-n) (1-\sec (c+d x))^2}+\frac{a^2 (n+3) \sec ^2(c+d x) (a \sec (c+d x)+a)^{n-2}}{4 d (1-n) (1-\sec (c+d x))^2}",1,"(a^2*(12 + 9*n + n^2)*Hypergeometric2F1[1, -2 + n, -1 + n, (1 + Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(-2 + n))/(16*d*(2 - n)) + (a^2*(3 + n)*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(-2 + n))/(4*d*(1 - n)*(1 - Sec[c + d*x])^2) - (a^2*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(-2 + n))/(d*(1 - n)*(1 - Sec[c + d*x])^2) - (a^2*(a + a*Sec[c + d*x])^(-2 + n)*(12 + 4*n - 7*n^2 - n^3 - 2*(1 - n)*(6 + n)*Sec[c + d*x]))/(8*d*(2 - 3*n + n^2)*(1 - Sec[c + d*x]))","A",5,5,21,0.2381,1,"{3873, 100, 149, 146, 68}"
152,1,230,0,0.6678581,"\int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^4,x]","\frac{2^{n+\frac{1}{2}} \sin (c+d x) \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};n-4,\frac{1}{2}-n;\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d}-\frac{\cot (c+d x) (n-n \cos (c+d x)) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},\frac{1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{1-\cos (c+d x)}}-\frac{\sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d}","\frac{2^{n+\frac{1}{2}} \sin (c+d x) \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};n-4,\frac{1}{2}-n;\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d}-\frac{\cot (c+d x) (n-n \cos (c+d x)) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},\frac{1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{1-\cos (c+d x)}}-\frac{\sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d}",1,"-((AppellF1[1 - n, -1/2, 1/2 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 + Cos[c + d*x])^(1/2 - n)*(n - n*Cos[c + d*x])*Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sqrt[1 - Cos[c + d*x]])) - (Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/d + (2^(1/2 + n)*AppellF1[1/2, -4 + n, 1/2 - n, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*Cos[c + d*x]^n*(1 + Cos[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/d","A",11,9,21,0.4286,1,"{3876, 2881, 2787, 2786, 2785, 133, 3046, 3008, 135}"
153,1,95,0,0.3530768,"\int (a+a \sec (c+d x))^n \sin ^2(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^2,x]","-\frac{\sqrt{1-\cos (c+d x)} \cot (c+d x) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},-n-\frac{1}{2};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}","-\frac{\sqrt{1-\cos (c+d x)} \cot (c+d x) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},-n-\frac{1}{2};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}",1,"-((AppellF1[1 - n, -1/2, -1/2 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(1/2 - n)*Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/(d*(1 - n)))","A",6,5,21,0.2381,1,"{3876, 2874, 3008, 135, 133}"
154,1,98,0,0.1322629,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]","\frac{2^{n-\frac{1}{2}} n \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{3}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d}-\frac{\cot (c+d x) (a \sec (c+d x)+a)^n}{d}","\frac{2^{n-\frac{1}{2}} n \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{3}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d}-\frac{\cot (c+d x) (a \sec (c+d x)+a)^n}{d}",1,"-((Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/d) + (2^(-1/2 + n)*n*Hypergeometric2F1[1/2, 3/2 - n, 3/2, (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d","A",4,4,21,0.1905,1,"{3875, 3828, 3827, 69}"
155,1,349,0,0.5414322,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]","-\frac{a^3 (4-n) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d \left(4 n^2-8 n+3\right) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)}-\frac{a^4 \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)^2}+\frac{n \left(-n^2-3 n+7\right) \sin (c+d x) \cos (c+d x) \left(\frac{\cos (c+d x)+1}{1-\cos (c+d x)}\right)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(-n-\frac{1}{2},1-n;2-n;-\frac{2 \cos (c+d x)}{1-\cos (c+d x)}\right)}{d (1-2 n) (3-2 n) (1-n) (2 n+1) (1-\cos (c+d x))^2}+\frac{\left(n^2-n+2\right) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) \left(1-4 n^2\right) (1-\cos (c+d x))^2}","-\frac{a^3 (4-n) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d \left(4 n^2-8 n+3\right) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)}-\frac{a^4 \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)^2}+\frac{n \left(-n^2-3 n+7\right) \sin (c+d x) \cos (c+d x) \left(\frac{\cos (c+d x)+1}{1-\cos (c+d x)}\right)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(-n-\frac{1}{2},1-n;2-n;-\frac{2 \cos (c+d x)}{1-\cos (c+d x)}\right)}{d (1-2 n) (3-2 n) (1-n) (2 n+1) (1-\cos (c+d x))^2}+\frac{\left(n^2-n+2\right) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) \left(1-4 n^2\right) (1-\cos (c+d x))^2}",1,"((2 - n + n^2)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 - 4*n^2)*(1 - Cos[c + d*x])^2) - (a^4*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])^2) - (a^3*(4 - n)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 8*n + 4*n^2)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])) + (n*(7 - 3*n - n^2)*Cos[c + d*x]*((1 + Cos[c + d*x])/(1 - Cos[c + d*x]))^(-1/2 - n)*Hypergeometric2F1[-1/2 - n, 1 - n, 2 - n, (-2*Cos[c + d*x])/(1 - Cos[c + d*x])]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 - 2*n)*(1 - n)*(1 + 2*n)*(1 - Cos[c + d*x])^2)","A",7,6,21,0.2857,1,"{3876, 2883, 129, 155, 12, 132}"
156,1,105,0,0.2617433,"\int (a+a \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^(3/2),x]","-\frac{\sqrt{\sin (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{4}} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{4},-n-\frac{1}{4};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt[4]{1-\cos (c+d x)}}","-\frac{\sqrt{\sin (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{4}} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{4},-n-\frac{1}{4};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt[4]{1-\cos (c+d x)}}",1,"-((AppellF1[1 - n, -1/4, -1/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*Cos[c + d*x]*(1 + Cos[c + d*x])^(-1/4 - n)*(a + a*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]])/(d*(1 - n)*(1 - Cos[c + d*x])^(1/4)))","A",5,4,23,0.1739,1,"{3876, 2886, 135, 133}"
157,1,105,0,0.2723983,"\int (a+a \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]],x]","-\frac{\sqrt[4]{1-\cos (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{\frac{1}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{1}{4},\frac{1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{\sin (c+d x)}}","-\frac{\sqrt[4]{1-\cos (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{\frac{1}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{1}{4},\frac{1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{\sin (c+d x)}}",1,"-((AppellF1[1 - n, 1/4, 1/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(1/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(1/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sqrt[Sin[c + d*x]]))","A",5,4,23,0.1739,1,"{3876, 2886, 135, 133}"
158,1,105,0,0.2495399,"\int \frac{(a+a \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]],x]","-\frac{(1-\cos (c+d x))^{3/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{3}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{3}{4},\frac{3}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{3}{2}}(c+d x)}","-\frac{(1-\cos (c+d x))^{3/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{3}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{3}{4},\frac{3}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{3}{2}}(c+d x)}",1,"-((AppellF1[1 - n, 3/4, 3/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(3/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(3/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sin[c + d*x]^(3/2)))","A",5,4,23,0.1739,1,"{3876, 2886, 135, 133}"
159,1,105,0,0.2665058,"\int \frac{(a+a \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + a*Sec[c + d*x])^n/Sin[c + d*x]^(3/2),x]","-\frac{(1-\cos (c+d x))^{5/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{5}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{5}{4},\frac{5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{5}{2}}(c+d x)}","-\frac{(1-\cos (c+d x))^{5/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{5}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{5}{4},\frac{5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{5}{2}}(c+d x)}",1,"-((AppellF1[1 - n, 5/4, 5/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(5/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(5/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sin[c + d*x]^(5/2)))","A",5,4,23,0.1739,1,"{3876, 2886, 135, 133}"
160,1,119,0,0.1095384,"\int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^7,x]","\frac{a \cos ^7(c+d x)}{7 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^6(c+d x)}{6 d}-\frac{3 b \cos ^4(c+d x)}{4 d}+\frac{3 b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}","\frac{a \cos ^7(c+d x)}{7 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^6(c+d x)}{6 d}-\frac{3 b \cos ^4(c+d x)}{4 d}+\frac{3 b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (3*b*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*b*Cos[c + d*x]^4)/(4*d) - (3*a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (b*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2837, 12, 766}"
161,1,87,0,0.0981964,"\int (a+b \sec (c+d x)) \sin ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^5,x]","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}+\frac{b \cos ^2(c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}+\frac{b \cos ^2(c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (b*Cos[c + d*x]^2)/d + (2*a*Cos[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]^4)/(4*d) - (a*Cos[c + d*x]^5)/(5*d) - (b*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2837, 12, 766}"
162,1,58,0,0.0857308,"\int (a+b \sec (c+d x)) \sin ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^3,x]","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) + (b*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/(3*d) - (b*Log[Cos[c + d*x]])/d","A",5,4,19,0.2105,1,"{3872, 2837, 12, 766}"
163,1,26,0,0.0329812,"\int (a+b \sec (c+d x)) \sin (c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x],x]","-\frac{a \cos (c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) - (b*Log[Cos[c + d*x]])/d","A",4,3,17,0.1765,1,"{3872, 2721, 43}"
164,1,26,0,0.0725366,"\int \csc (c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{b \log (\tan (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{b \log (\tan (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) + (b*Log[Tan[c + d*x]])/d","A",5,5,17,0.2941,1,"{3872, 2834, 2620, 29, 3770}"
165,1,64,0,0.1035205,"\int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^3*(a + b*Sec[c + d*x]),x]","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"-(a*ArcTanh[Cos[c + d*x]])/(2*d) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (b*Log[Tan[c + d*x]])/d","A",7,6,19,0.3158,1,"{3872, 2834, 2620, 14, 3768, 3770}"
166,1,100,0,0.1241943,"\int \csc ^5(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^5*(a + b*Sec[c + d*x]),x]","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(8*d) - (b*Cot[c + d*x]^2)/d - (b*Cot[c + d*x]^4)/(4*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (b*Log[Tan[c + d*x]])/d","A",9,7,19,0.3684,1,"{3872, 2834, 2620, 266, 43, 3768, 3770}"
167,1,140,0,0.1443514,"\int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^7*(a + b*Sec[c + d*x]),x]","-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{3 b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{3 b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (3*b*Cot[c + d*x]^2)/(2*d) - (3*b*Cot[c + d*x]^4)/(4*d) - (b*Cot[c + d*x]^6)/(6*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (b*Log[Tan[c + d*x]])/d","A",10,7,19,0.3684,1,"{3872, 2834, 2620, 266, 43, 3768, 3770}"
168,1,127,0,0.1284007,"\int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^6,x]","-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(5*a*x)/16 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (b*Sin[c + d*x]^3)/(3*d) - (5*a*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (b*Sin[c + d*x]^5)/(5*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",10,7,19,0.3684,1,"{3872, 2838, 2592, 302, 206, 2635, 8}"
169,1,89,0,0.1106733,"\int (a+b \sec (c+d x)) \sin ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^4,x]","-\frac{a \sin ^3(c+d x) \cos (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{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin ^3(c+d x) \cos (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{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(3*a*x)/8 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Sin[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",9,7,19,0.3684,1,"{3872, 2838, 2592, 302, 206, 2635, 8}"
170,1,51,0,0.0825193,"\int (a+b \sec (c+d x)) \sin ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])*Sin[c + d*x]^2,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{b \tanh ^{-1}(\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}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*x)/2 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",7,7,19,0.3684,1,"{3872, 2838, 2592, 321, 206, 2635, 8}"
171,1,37,0,0.0959633,"\int \csc ^2(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^2*(a + b*Sec[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot (c+d x)}{d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (b*Csc[c + d*x])/d","A",7,7,19,0.3684,1,"{3872, 2838, 2621, 321, 207, 3767, 8}"
172,1,69,0,0.1046646,"\int \csc ^4(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^4*(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) - (b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
173,1,101,0,0.1108907,"\int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx","Int[Csc[c + d*x]^6*(a + b*Sec[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (2*a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)","A",8,6,19,0.3158,1,"{3872, 2838, 2621, 302, 207, 3767}"
174,1,124,0,0.1956424,"\int (a+b \sec (c+d x))^2 \sin ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^5,x]","\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a b \cos ^4(c+d x)}{2 d}+\frac{2 a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}","\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a b \cos ^4(c+d x)}{2 d}+\frac{2 a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-(((a^2 - 2*b^2)*Cos[c + d*x])/d) + (2*a*b*Cos[c + d*x]^2)/d + ((2*a^2 - b^2)*Cos[c + d*x]^3)/(3*d) - (a*b*Cos[c + d*x]^4)/(2*d) - (a^2*Cos[c + d*x]^5)/(5*d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
175,1,80,0,0.1447945,"\int (a+b \sec (c+d x))^2 \sin ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^3,x]","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-(((a^2 - b^2)*Cos[c + d*x])/d) + (a*b*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d","A",5,4,21,0.1905,1,"{3872, 2837, 12, 894}"
176,1,42,0,0.0775606,"\int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x],x]","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-((a^2*Cos[c + d*x])/d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
177,1,74,0,0.1802781,"\int \csc (c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]*(a + b*Sec[c + d*x])^2,x]","-\frac{(a-b)^2 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{(a-b)^2 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"((a + b)^2*Log[1 - Cos[c + d*x]])/(2*d) - (2*a*b*Log[Cos[c + d*x]])/d - ((a - b)^2*Log[1 + Cos[c + d*x]])/(2*d) + (b^2*Sec[c + d*x])/d","A",5,4,19,0.2105,1,"{3872, 2837, 12, 1802}"
178,1,114,0,0.2941164,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^2(c+d x) \left(\left(a^2+b^2\right) \cos (c+d x)+2 a b\right)}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-3 b) (a-b) \log (\cos (c+d x)+1)}{4 d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{\csc ^2(c+d x) \left(\left(a^2+b^2\right) \cos (c+d x)+2 a b\right)}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-3 b) (a-b) \log (\cos (c+d x)+1)}{4 d}+\frac{b^2 \sec (c+d x)}{d}",1,"-((2*a*b + (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*d) + ((a + b)*(a + 3*b)*Log[1 - Cos[c + d*x]])/(4*d) - (2*a*b*Log[Cos[c + d*x]])/d - ((a - 3*b)*(a - b)*Log[1 + Cos[c + d*x]])/(4*d) + (b^2*Sec[c + d*x])/d","A",6,5,21,0.2381,1,"{3872, 2837, 12, 1805, 1802}"
179,1,175,0,0.4612753,"\int (a+b \sec (c+d x))^2 \sin ^6(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^6,x]","\frac{\left(13 a^2-6 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(11 a^2-18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5}{16} x \left(a^2-6 b^2\right)-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","\frac{\left(13 a^2-6 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(11 a^2-18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5}{16} x \left(a^2-6 b^2\right)-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(5*(a^2 - 6*b^2)*x)/16 + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - ((11*a^2 - 18*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((13*a^2 - 6*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*b*Sin[c + d*x]^3)/(3*d) - (2*a*b*Sin[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x])/d","A",12,10,21,0.4762,1,"{3872, 2911, 2592, 302, 206, 455, 1814, 1157, 388, 203}"
180,1,178,0,0.5564683,"\int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^4,x]","-\frac{b \left(28 a^2+b^2\right) \sin (c+d x)}{6 a d}-\frac{\left(12 a^2+b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{12 a b d}-\frac{\left(39 a^2+2 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{3}{8} x \left(a^2-4 b^2\right)+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac{\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}","-\frac{b \left(28 a^2+b^2\right) \sin (c+d x)}{6 a d}-\frac{\left(12 a^2+b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{12 a b d}-\frac{\left(39 a^2+2 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{3}{8} x \left(a^2-4 b^2\right)+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac{\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}",1,"(3*(a^2 - 4*b^2)*x)/8 + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (b*(28*a^2 + b^2)*Sin[c + d*x])/(6*a*d) - ((39*a^2 + 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - ((12*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*a*b*d) + ((b + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d) + ((b + a*Cos[c + d*x])^3*Tan[c + d*x])/(b*d)","A",7,7,21,0.3333,1,"{3872, 2894, 3049, 3033, 3023, 2735, 3770}"
181,1,77,0,0.1312285,"\int (a+b \sec (c+d x))^2 \sin ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^2,x]","-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^2 x}{2}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}-b^2 x","-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^2 x}{2}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}-b^2 x",1,"(a^2*x)/2 - b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d","A",10,8,21,0.3810,1,"{3872, 2722, 2635, 8, 2592, 321, 206, 3473}"
182,1,59,0,0.4140071,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + b^2)*Cot[c + d*x])/d - (2*a*b*Csc[c + d*x])/d + (b^2*Tan[c + d*x])/d","A",8,6,21,0.2857,1,"{3872, 2911, 2621, 321, 207, 14}"
183,1,100,0,0.3219169,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{\left(a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + 2*b^2)*Cot[c + d*x])/d - ((a^2 + b^2)*Cot[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x])/d","A",9,6,21,0.2857,1,"{3872, 2911, 2621, 302, 207, 448}"
184,1,143,0,0.407985,"\int \csc ^6(c+d x) (a+b \sec (c+d x))^2 \, dx","Int[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot ^5(c+d x)}{5 d}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{\left(a^2+b^2\right) \cot ^5(c+d x)}{5 d}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + 3*b^2)*Cot[c + d*x])/d - ((2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - ((a^2 + b^2)*Cot[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x])/d","A",9,6,21,0.2857,1,"{3872, 2911, 2621, 302, 207, 448}"
185,1,170,0,0.2551286,"\int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^5,x]","\frac{a \left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 d}+\frac{b \left(6 a^2-b^2\right) \cos ^2(c+d x)}{2 d}-\frac{a \left(a^2-6 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}-\frac{3 a^2 b \cos ^4(c+d x)}{4 d}-\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","\frac{a \left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 d}+\frac{b \left(6 a^2-b^2\right) \cos ^2(c+d x)}{2 d}-\frac{a \left(a^2-6 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}-\frac{3 a^2 b \cos ^4(c+d x)}{4 d}-\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"-((a*(a^2 - 6*b^2)*Cos[c + d*x])/d) + (b*(6*a^2 - b^2)*Cos[c + d*x]^2)/(2*d) + (a*(2*a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*d) - (3*a^2*b*Cos[c + d*x]^4)/(4*d) - (a^3*Cos[c + d*x]^5)/(5*d) - (b*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
186,1,116,0,0.1281578,"\int (a+b \sec (c+d x))^3 \sin ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^3,x]","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"-((a*(a^2 - 3*b^2)*Cos[c + d*x])/d) + (3*a^2*b*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",4,3,21,0.1429,1,"{3872, 2721, 894}"
187,1,64,0,0.1011725,"\int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x],x]","-\frac{3 a^2 b \log (\cos (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","-\frac{3 a^2 b \log (\cos (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"-((a^3*Cos[c + d*x])/d) - (3*a^2*b*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
188,1,102,0,0.2192127,"\int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]*(a + b*Sec[c + d*x])^3,x]","-\frac{b \left(3 a^2+b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{(a-b)^3 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^3 \log (1-\cos (c+d x))}{2 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","-\frac{b \left(3 a^2+b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{(a-b)^3 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^3 \log (1-\cos (c+d x))}{2 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"((a + b)^3*Log[1 - Cos[c + d*x]])/(2*d) - (b*(3*a^2 + b^2)*Log[Cos[c + d*x]])/d - ((a - b)^3*Log[1 + Cos[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",5,4,19,0.2105,1,"{3872, 2837, 12, 1802}"
189,1,162,0,0.3489481,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","-\frac{b \left(3 a^2+2 b^2\right) \log (\cos (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x) \left(a \left(\frac{3 b^2}{a^2}+1\right) \cos (c+d x)+b \left(\frac{b^2}{a^2}+3\right)\right)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{(a+b)^2 (a+4 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-4 b) (a-b)^2 \log (\cos (c+d x)+1)}{4 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","-\frac{b \left(3 a^2+2 b^2\right) \log (\cos (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x) \left(a \left(\frac{3 b^2}{a^2}+1\right) \cos (c+d x)+b \left(\frac{b^2}{a^2}+3\right)\right)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{(a+b)^2 (a+4 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-4 b) (a-b)^2 \log (\cos (c+d x)+1)}{4 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"-(a^2*(b*(3 + b^2/a^2) + a*(1 + (3*b^2)/a^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*d) + ((a + b)^2*(a + 4*b)*Log[1 - Cos[c + d*x]])/(4*d) - (b*(3*a^2 + 2*b^2)*Log[Cos[c + d*x]])/d - ((a - 4*b)*(a - b)^2*Log[1 + Cos[c + d*x]])/(4*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",6,5,21,0.2381,1,"{3872, 2837, 12, 1805, 1802}"
190,1,299,0,0.3359316,"\int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^6,x]","-\frac{3 a^2 b \sin ^5(c+d x)}{5 d}-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}+\frac{45 a b^2 \tan (c+d x)}{8 d}-\frac{3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac{15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac{45}{8} a b^2 x+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}","-\frac{3 a^2 b \sin ^5(c+d x)}{5 d}-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}+\frac{45 a b^2 \tan (c+d x)}{8 d}-\frac{3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac{15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac{45}{8} a b^2 x+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(5*a^3*x)/16 - (45*a*b^2*x)/8 + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Sin[c + d*x])/d + (5*b^3*Sin[c + d*x])/(2*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a^2*b*Sin[c + d*x]^3)/d + (5*b^3*Sin[c + d*x]^3)/(6*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (3*a^2*b*Sin[c + d*x]^5)/(5*d) - (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d) + (45*a*b^2*Tan[c + d*x])/(8*d) - (15*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(8*d) - (3*a*b^2*Sin[c + d*x]^4*Tan[c + d*x])/(4*d) + (b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(2*d)","A",21,11,21,0.5238,1,"{3872, 2912, 2635, 8, 2592, 302, 206, 2591, 288, 321, 203}"
191,1,236,0,0.7480798,"\int (a+b \sec (c+d x))^3 \sin ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^4,x]","-\frac{b \left(17 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{3 b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^3}{4 b^2 d}-\frac{\left(6 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{4 b d}-\frac{a \left(21 a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2-12 b^2\right)+\frac{a \tan (c+d x) (a \cos (c+d x)+b)^4}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^4}{2 b d}","-\frac{b \left(17 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{3 b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^3}{4 b^2 d}-\frac{\left(6 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{4 b d}-\frac{a \left(21 a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2-12 b^2\right)+\frac{a \tan (c+d x) (a \cos (c+d x)+b)^4}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^4}{2 b d}",1,"(3*a*(a^2 - 12*b^2)*x)/8 + (3*b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(17*a^2 - b^2)*Sin[c + d*x])/(2*d) - (a*(21*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - ((6*a^2 - b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(4*b*d) - ((4*a^2 - b^2)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*b^2*d) + (a*(b + a*Cos[c + d*x])^4*Tan[c + d*x])/(b^2*d) + ((b + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",8,7,21,0.3333,1,"{3872, 2893, 3049, 3033, 3023, 2735, 3770}"
192,1,138,0,0.5038205,"\int (a+b \sec (c+d x))^3 \sin ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^2,x]","\frac{b \left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} a x \left(a^2-6 b^2\right)-\frac{15 a^2 b \sin (c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a \tan (c+d x) (a \cos (c+d x)+b)^2}{2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^3}{2 d}","\frac{b \left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} a x \left(a^2-6 b^2\right)-\frac{15 a^2 b \sin (c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a \tan (c+d x) (a \cos (c+d x)+b)^2}{2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^3}{2 d}",1,"(a*(a^2 - 6*b^2)*x)/2 + (b*(6*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (15*a^2*b*Sin[c + d*x])/(2*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a*(b + a*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",8,8,21,0.3810,1,"{3872, 2889, 3048, 3047, 3033, 3023, 2735, 3770}"
193,1,133,0,0.2725076,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot (c+d x)}{d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}","-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot (c+d x)}{d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}",1,"(3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (3*a*b^2*Cot[c + d*x])/d - (3*a^2*b*Csc[c + d*x])/d - (3*b^3*Csc[c + d*x])/(2*d) + (b^3*Csc[c + d*x]*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d","A",15,10,21,0.4762,1,"{3872, 2912, 3767, 8, 2621, 321, 207, 2620, 14, 288}"
194,1,205,0,0.2912417,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^3,x]","-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{a b^2 \cot ^3(c+d x)}{d}-\frac{6 a b^2 \cot (c+d x)}{d}-\frac{5 b^3 \csc ^3(c+d x)}{6 d}-\frac{5 b^3 \csc (c+d x)}{2 d}+\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{2 d}","-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{a b^2 \cot ^3(c+d x)}{d}-\frac{6 a b^2 \cot (c+d x)}{d}-\frac{5 b^3 \csc ^3(c+d x)}{6 d}-\frac{5 b^3 \csc (c+d x)}{2 d}+\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{2 d}",1,"(3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (6*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (a*b^2*Cot[c + d*x]^3)/d - (3*a^2*b*Csc[c + d*x])/d - (5*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (5*b^3*Csc[c + d*x]^3)/(6*d) + (b^3*Csc[c + d*x]^3*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d","A",17,9,21,0.4286,1,"{3872, 2912, 3767, 2621, 302, 207, 2620, 270, 288}"
195,1,279,0,0.3162174,"\int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx","Int[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]","-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{7 b^3 \csc (c+d x)}{2 d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}","-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{7 b^3 \csc (c+d x)}{2 d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}",1,"(3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (7*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (3*a*b^2*Cot[c + d*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a*b^2*Cot[c + d*x]^5)/(5*d) - (3*a^2*b*Csc[c + d*x])/d - (7*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (7*b^3*Csc[c + d*x]^3)/(6*d) - (3*a^2*b*Csc[c + d*x]^5)/(5*d) - (7*b^3*Csc[c + d*x]^5)/(10*d) + (b^3*Csc[c + d*x]^5*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d","A",17,9,21,0.4286,1,"{3872, 2912, 3767, 2621, 302, 207, 2620, 270, 288}"
196,1,223,0,0.2509375,"\int \frac{\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]","-\frac{\left(3 a^2-b^2\right) \cos ^5(c+d x)}{5 a^3 d}+\frac{b \left(3 a^2-b^2\right) \cos ^4(c+d x)}{4 a^4 d}+\frac{\left(-3 a^2 b^2+3 a^4+b^4\right) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left(-3 a^2 b^2+3 a^4+b^4\right) \cos ^2(c+d x)}{2 a^6 d}-\frac{\left(a^2-b^2\right)^3 \cos (c+d x)}{a^7 d}+\frac{b \left(a^2-b^2\right)^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac{b \cos ^6(c+d x)}{6 a^2 d}+\frac{\cos ^7(c+d x)}{7 a d}","-\frac{\left(3 a^2-b^2\right) \cos ^5(c+d x)}{5 a^3 d}+\frac{b \left(3 a^2-b^2\right) \cos ^4(c+d x)}{4 a^4 d}+\frac{\left(-3 a^2 b^2+3 a^4+b^4\right) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left(-3 a^2 b^2+3 a^4+b^4\right) \cos ^2(c+d x)}{2 a^6 d}-\frac{\left(a^2-b^2\right)^3 \cos (c+d x)}{a^7 d}+\frac{b \left(a^2-b^2\right)^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac{b \cos ^6(c+d x)}{6 a^2 d}+\frac{\cos ^7(c+d x)}{7 a d}",1,"-(((a^2 - b^2)^3*Cos[c + d*x])/(a^7*d)) - (b*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^2)/(2*a^6*d) + ((3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^3)/(3*a^5*d) + (b*(3*a^2 - b^2)*Cos[c + d*x]^4)/(4*a^4*d) - ((3*a^2 - b^2)*Cos[c + d*x]^5)/(5*a^3*d) - (b*Cos[c + d*x]^6)/(6*a^2*d) + Cos[c + d*x]^7/(7*a*d) + (b*(a^2 - b^2)^3*Log[b + a*Cos[c + d*x]])/(a^8*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 772}"
197,1,152,0,0.1941318,"\int \frac{\sin ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 a^3 d}-\frac{b \left(2 a^2-b^2\right) \cos ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^5 d}+\frac{b \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^6 d}+\frac{b \cos ^4(c+d x)}{4 a^2 d}-\frac{\cos ^5(c+d x)}{5 a d}","\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 a^3 d}-\frac{b \left(2 a^2-b^2\right) \cos ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^5 d}+\frac{b \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^6 d}+\frac{b \cos ^4(c+d x)}{4 a^2 d}-\frac{\cos ^5(c+d x)}{5 a d}",1,"-(((a^2 - b^2)^2*Cos[c + d*x])/(a^5*d)) - (b*(2*a^2 - b^2)*Cos[c + d*x]^2)/(2*a^4*d) + ((2*a^2 - b^2)*Cos[c + d*x]^3)/(3*a^3*d) + (b*Cos[c + d*x]^4)/(4*a^2*d) - Cos[c + d*x]^5/(5*a*d) + (b*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]])/(a^6*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 772}"
198,1,89,0,0.1557301,"\int \frac{\sin ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^3/(a + b*Sec[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a^3 d}+\frac{b \left(a^2-b^2\right) \log (a \cos (c+d x)+b)}{a^4 d}-\frac{b \cos ^2(c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x)}{3 a d}","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a^3 d}+\frac{b \left(a^2-b^2\right) \log (a \cos (c+d x)+b)}{a^4 d}-\frac{b \cos ^2(c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"-(((a^2 - b^2)*Cos[c + d*x])/(a^3*d)) - (b*Cos[c + d*x]^2)/(2*a^2*d) + Cos[c + d*x]^3/(3*a*d) + (b*(a^2 - b^2)*Log[b + a*Cos[c + d*x]])/(a^4*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 772}"
199,1,34,0,0.0766032,"\int \frac{\sin (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d}","\frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d}",1,"-(Cos[c + d*x]/(a*d)) + (b*Log[b + a*Cos[c + d*x]])/(a^2*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
200,1,74,0,0.1053448,"\int \frac{\csc (c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}","\frac{b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)","A",4,3,19,0.1579,1,"{3872, 2721, 801}"
201,1,116,0,0.21299,"\int \frac{\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{a^2 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{a \log (1-\cos (c+d x))}{4 d (a+b)^2}-\frac{a \log (\cos (c+d x)+1)}{4 d (a-b)^2}","\frac{a^2 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{a \log (1-\cos (c+d x))}{4 d (a+b)^2}-\frac{a \log (\cos (c+d x)+1)}{4 d (a-b)^2}",1,"((b - a*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d) + (a*Log[1 - Cos[c + d*x]])/(4*(a + b)^2*d) - (a*Log[1 + Cos[c + d*x]])/(4*(a - b)^2*d) + (a^2*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)","A",6,5,21,0.2381,1,"{3872, 2837, 12, 823, 801}"
202,1,179,0,0.3013899,"\int \frac{\csc ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{a^4 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\csc ^4(c+d x) (b-a \cos (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\csc ^2(c+d x) \left(4 a^2 b-a \left(3 a^2+b^2\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a (3 a+b) \log (1-\cos (c+d x))}{16 d (a+b)^3}-\frac{a (3 a-b) \log (\cos (c+d x)+1)}{16 d (a-b)^3}","\frac{a^4 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\csc ^4(c+d x) (b-a \cos (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\csc ^2(c+d x) \left(4 a^2 b-a \left(3 a^2+b^2\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a (3 a+b) \log (1-\cos (c+d x))}{16 d (a+b)^3}-\frac{a (3 a-b) \log (\cos (c+d x)+1)}{16 d (a-b)^3}",1,"((4*a^2*b - a*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)*d) + (a*(3*a + b)*Log[1 - Cos[c + d*x]])/(16*(a + b)^3*d) - (a*(3*a - b)*Log[1 + Cos[c + d*x]])/(16*(a - b)^3*d) + (a^4*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)","A",7,5,21,0.2381,1,"{3872, 2837, 12, 823, 801}"
203,1,230,0,0.6075887,"\int \frac{\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x]),x]","\frac{\sin ^3(c+d x) \left(8 b \left(a^2-b^2\right)-a \left(5 a^2-6 b^2\right) \cos (c+d x)\right)}{24 a^4 d}+\frac{\sin (c+d x) \left(16 b \left(a^2-b^2\right)^2-a \left(-14 a^2 b^2+5 a^4+8 b^4\right) \cos (c+d x)\right)}{16 a^6 d}+\frac{x \left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right)}{16 a^7}+\frac{\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac{2 b (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d}","\frac{\sin ^3(c+d x) \left(8 b \left(a^2-b^2\right)-a \left(5 a^2-6 b^2\right) \cos (c+d x)\right)}{24 a^4 d}+\frac{\sin (c+d x) \left(16 b \left(a^2-b^2\right)^2-a \left(-14 a^2 b^2+5 a^4+8 b^4\right) \cos (c+d x)\right)}{16 a^6 d}+\frac{x \left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right)}{16 a^7}+\frac{\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}-\frac{2 b (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d}",1,"((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*x)/(16*a^7) - (2*(a - b)^(5/2)*b*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^7*d) + ((16*b*(a^2 - b^2)^2 - a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*Cos[c + d*x])*Sin[c + d*x])/(16*a^6*d) + ((8*b*(a^2 - b^2) - a*(5*a^2 - 6*b^2)*Cos[c + d*x])*Sin[c + d*x]^3)/(24*a^4*d) + ((6*b - 5*a*Cos[c + d*x])*Sin[c + d*x]^5)/(30*a^2*d)","A",7,5,21,0.2381,1,"{3872, 2865, 2735, 2659, 208}"
204,1,161,0,0.3807181,"\int \frac{\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{\sin (c+d x) \left(8 b \left(a^2-b^2\right)-a \left(3 a^2-4 b^2\right) \cos (c+d x)\right)}{8 a^4 d}+\frac{x \left(-12 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}+\frac{\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d}","\frac{\sin (c+d x) \left(8 b \left(a^2-b^2\right)-a \left(3 a^2-4 b^2\right) \cos (c+d x)\right)}{8 a^4 d}+\frac{x \left(-12 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}+\frac{\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d}",1,"((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*d) + ((8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/(8*a^4*d) + ((4*b - 3*a*Cos[c + d*x])*Sin[c + d*x]^3)/(12*a^2*d)","A",6,5,21,0.2381,1,"{3872, 2865, 2735, 2659, 208}"
205,1,100,0,0.2070728,"\int \frac{\sin ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Sin[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{x \left(a^2-2 b^2\right)}{2 a^3}+\frac{\sin (c+d x) (2 b-a \cos (c+d x))}{2 a^2 d}-\frac{2 b \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d}","\frac{x \left(a^2-2 b^2\right)}{2 a^3}+\frac{\sin (c+d x) (2 b-a \cos (c+d x))}{2 a^2 d}-\frac{2 b \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d}",1,"((a^2 - 2*b^2)*x)/(2*a^3) - (2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*d) + ((2*b - a*Cos[c + d*x])*Sin[c + d*x])/(2*a^2*d)","A",5,5,21,0.2381,1,"{3872, 2865, 2735, 2659, 208}"
206,1,84,0,0.1487118,"\int \frac{\csc ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{\csc (c+d x) (b-a \cos (c+d x))}{d \left(a^2-b^2\right)}-\frac{2 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)^{3/2} (a+b)^{3/2}}","\frac{\csc (c+d x) (b-a \cos (c+d x))}{d \left(a^2-b^2\right)}-\frac{2 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)^{3/2} (a+b)^{3/2}}",1,"(-2*a*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)*d)","A",5,5,21,0.2381,1,"{3872, 2866, 12, 2659, 208}"
207,1,140,0,0.306471,"\int \frac{\csc ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{\csc ^3(c+d x) (b-a \cos (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{\csc (c+d x) \left(3 a^2 b-a \left(2 a^2+b^2\right) \cos (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}-\frac{2 a^3 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{\csc ^3(c+d x) (b-a \cos (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{\csc (c+d x) \left(3 a^2 b-a \left(2 a^2+b^2\right) \cos (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}-\frac{2 a^3 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,"(-2*a^3*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) + ((3*a^2*b - a*(2*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x])/(3*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",6,5,21,0.2381,1,"{3872, 2866, 12, 2659, 208}"
208,1,201,0,0.519973,"\int \frac{\csc ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Int[Csc[c + d*x]^6/(a + b*Sec[c + d*x]),x]","\frac{\csc ^5(c+d x) (b-a \cos (c+d x))}{5 d \left(a^2-b^2\right)}+\frac{\csc ^3(c+d x) \left(5 a^2 b-a \left(4 a^2+b^2\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^2}+\frac{\csc (c+d x) \left(15 a^4 b-a \left(9 a^2 b^2+8 a^4-2 b^4\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}-\frac{2 a^5 b \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{\csc ^5(c+d x) (b-a \cos (c+d x))}{5 d \left(a^2-b^2\right)}+\frac{\csc ^3(c+d x) \left(5 a^2 b-a \left(4 a^2+b^2\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^2}+\frac{\csc (c+d x) \left(15 a^4 b-a \left(9 a^2 b^2+8 a^4-2 b^4\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}-\frac{2 a^5 b \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}}",1,"(-2*a^5*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) + ((15*a^4*b - a*(8*a^4 + 9*a^2*b^2 - 2*b^4)*Cos[c + d*x])*Csc[c + d*x])/(15*(a^2 - b^2)^3*d) + ((5*a^2*b - a*(4*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^3)/(15*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^5)/(5*(a^2 - b^2)*d)","A",7,5,21,0.2381,1,"{3872, 2866, 12, 2659, 208}"
209,1,267,0,0.3723329,"\int \frac{\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]","-\frac{3 \left(a^2-b^2\right) \cos ^5(c+d x)}{5 a^4 d}+\frac{b \left(3 a^2-2 b^2\right) \cos ^4(c+d x)}{2 a^5 d}+\frac{\left(-9 a^2 b^2+3 a^4+5 b^4\right) \cos ^3(c+d x)}{3 a^6 d}-\frac{3 b \left(a^2-b^2\right)^2 \cos ^2(c+d x)}{a^7 d}-\frac{\left(a^2-7 b^2\right) \left(a^2-b^2\right)^2 \cos (c+d x)}{a^8 d}+\frac{b^2 \left(a^2-b^2\right)^3}{a^9 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-4 b^2\right) \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}","-\frac{3 \left(a^2-b^2\right) \cos ^5(c+d x)}{5 a^4 d}+\frac{b \left(3 a^2-2 b^2\right) \cos ^4(c+d x)}{2 a^5 d}+\frac{\left(-9 a^2 b^2+3 a^4+5 b^4\right) \cos ^3(c+d x)}{3 a^6 d}-\frac{3 b \left(a^2-b^2\right)^2 \cos ^2(c+d x)}{a^7 d}-\frac{\left(a^2-7 b^2\right) \left(a^2-b^2\right)^2 \cos (c+d x)}{a^8 d}+\frac{b^2 \left(a^2-b^2\right)^3}{a^9 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-4 b^2\right) \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}",1,"-(((a^2 - 7*b^2)*(a^2 - b^2)^2*Cos[c + d*x])/(a^8*d)) - (3*b*(a^2 - b^2)^2*Cos[c + d*x]^2)/(a^7*d) + ((3*a^4 - 9*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(3*a^6*d) + (b*(3*a^2 - 2*b^2)*Cos[c + d*x]^4)/(2*a^5*d) - (3*(a^2 - b^2)*Cos[c + d*x]^5)/(5*a^4*d) - (b*Cos[c + d*x]^6)/(3*a^3*d) + Cos[c + d*x]^7/(7*a^2*d) + (b^2*(a^2 - b^2)^3)/(a^9*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]])/(a^9*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
210,1,194,0,0.2993517,"\int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^4 d}-\frac{2 b \left(a^2-b^2\right) \cos ^2(c+d x)}{a^5 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \cos (c+d x)}{a^6 d}+\frac{b^2 \left(a^2-b^2\right)^2}{a^7 d (a \cos (c+d x)+b)}+\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (a \cos (c+d x)+b)}{a^7 d}+\frac{b \cos ^4(c+d x)}{2 a^3 d}-\frac{\cos ^5(c+d x)}{5 a^2 d}","\frac{\left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^4 d}-\frac{2 b \left(a^2-b^2\right) \cos ^2(c+d x)}{a^5 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \cos (c+d x)}{a^6 d}+\frac{b^2 \left(a^2-b^2\right)^2}{a^7 d (a \cos (c+d x)+b)}+\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (a \cos (c+d x)+b)}{a^7 d}+\frac{b \cos ^4(c+d x)}{2 a^3 d}-\frac{\cos ^5(c+d x)}{5 a^2 d}",1,"-(((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x])/(a^6*d)) - (2*b*(a^2 - b^2)*Cos[c + d*x]^2)/(a^5*d) + ((2*a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*a^4*d) + (b*Cos[c + d*x]^4)/(2*a^3*d) - Cos[c + d*x]^5/(5*a^2*d) + (b^2*(a^2 - b^2)^2)/(a^7*d*(b + a*Cos[c + d*x])) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]])/(a^7*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
211,1,119,0,0.2283874,"\int \frac{\sin ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{a^4 d}+\frac{b^2 \left(a^2-b^2\right)}{a^5 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b)}{a^5 d}-\frac{b \cos ^2(c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}","-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{a^4 d}+\frac{b^2 \left(a^2-b^2\right)}{a^5 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b)}{a^5 d}-\frac{b \cos ^2(c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}",1,"-(((a^2 - 3*b^2)*Cos[c + d*x])/(a^4*d)) - (b*Cos[c + d*x]^2)/(a^3*d) + Cos[c + d*x]^3/(3*a^2*d) + (b^2*(a^2 - b^2))/(a^5*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 2*b^2)*Log[b + a*Cos[c + d*x]])/(a^5*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 894}"
212,1,57,0,0.1122594,"\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d}","\frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d}",1,"-(Cos[c + d*x]/(a^2*d)) + b^2/(a^3*d*(b + a*Cos[c + d*x])) + (2*b*Log[b + a*Cos[c + d*x]])/(a^3*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
213,1,109,0,0.2264323,"\int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{b^2}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2}","\frac{b^2}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2}",1,"b^2/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)^2*d) + (2*a*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)","A",5,4,19,0.2105,1,"{3872, 2837, 12, 1629}"
214,1,168,0,0.4328964,"\int \frac{\csc ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{a b^2}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 a b \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\csc ^2(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^2}+\frac{(a-b) \log (1-\cos (c+d x))}{4 d (a+b)^3}-\frac{(a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^3}","\frac{a b^2}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 a b \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\csc ^2(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^2}+\frac{(a-b) \log (1-\cos (c+d x))}{4 d (a+b)^3}-\frac{(a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^3}",1,"(a*b^2)/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x])) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^2*d) + ((a - b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^3*d) - ((a + b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^3*d) + (2*a*b*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)","A",6,5,21,0.2381,1,"{3872, 2837, 12, 1647, 1629}"
215,1,259,0,0.7411043,"\int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 b^2}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{\left(3 a^2-4 a b-b^2\right) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac{\left(3 a^2+4 a b-b^2\right) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac{2 a^3 b \left(a^2+2 b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^4(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) \left(8 a b \left(a^2+b^2\right)-\left(12 a^2 b^2+3 a^4+b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^3}","\frac{a^3 b^2}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{\left(3 a^2-4 a b-b^2\right) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac{\left(3 a^2+4 a b-b^2\right) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac{2 a^3 b \left(a^2+2 b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^4(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) \left(8 a b \left(a^2+b^2\right)-\left(12 a^2 b^2+3 a^4+b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^3}",1,"(a^3*b^2)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((8*a*b*(a^2 + b^2) - (3*a^4 + 12*a^2*b^2 + b^4)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^3*d) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^2*d) + ((3*a^2 - 4*a*b - b^2)*Log[1 - Cos[c + d*x]])/(16*(a + b)^4*d) - ((3*a^2 + 4*a*b - b^2)*Log[1 + Cos[c + d*x]])/(16*(a - b)^4*d) + (2*a^3*b*(a^2 + 2*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",7,5,21,0.2381,1,"{3872, 2837, 12, 1647, 1629}"
216,1,473,0,1.7129632,"\int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]","\frac{b \left(-170 a^2 b^2+61 a^4+105 b^4\right) \sin (c+d x)}{15 a^7 d}+\frac{\left(-20 a^2 b^2+5 a^4+14 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}-\frac{\left(-61 a^2 b^2+16 a^4+42 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left(-52 a^2 b^2+15 a^4+35 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}-\frac{\left(-86 a^2 b^2+27 a^4+56 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^6 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \left(2 a^2-7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^8 d}+\frac{x \left(-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right)}{16 a^8}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}+\frac{a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)}","\frac{b \left(-170 a^2 b^2+61 a^4+105 b^4\right) \sin (c+d x)}{15 a^7 d}+\frac{\left(-20 a^2 b^2+5 a^4+14 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}-\frac{\left(-61 a^2 b^2+16 a^4+42 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left(-52 a^2 b^2+15 a^4+35 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}-\frac{\left(-86 a^2 b^2+27 a^4+56 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^6 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \left(2 a^2-7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^8 d}+\frac{x \left(-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right)}{16 a^8}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}+\frac{a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)}",1,"((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*x)/(16*a^8) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*(2*a^2 - 7*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Sin[c + d*x])/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^4*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*b*d*(b + a*Cos[c + d*x])) + (a*Cos[c + d*x]^4*Sin[c + d*x])/(6*b^2*d*(b + a*Cos[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(10*a^3*b^2*d*(b + a*Cos[c + d*x])) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(30*a^2*d*(b + a*Cos[c + d*x])) - (Cos[c + d*x]^6*Sin[c + d*x])/(6*a*d*(b + a*Cos[c + d*x]))","A",10,8,21,0.3810,1,"{3872, 2896, 3047, 3049, 3023, 2735, 2659, 208}"
217,1,261,0,0.8252785,"\int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{b \left(11 a^2-15 b^2\right) \sin (c+d x)}{3 a^5 d}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac{\left(3 a^2-5 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^3 b d}-\frac{\left(13 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^4 d}-\frac{2 b \sqrt{a-b} \sqrt{a+b} \left(2 a^2-5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d}+\frac{x \left(-36 a^2 b^2+3 a^4+40 b^4\right)}{8 a^6}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}","\frac{b \left(11 a^2-15 b^2\right) \sin (c+d x)}{3 a^5 d}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac{\left(3 a^2-5 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^3 b d}-\frac{\left(13 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^4 d}-\frac{2 b \sqrt{a-b} \sqrt{a+b} \left(2 a^2-5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d}+\frac{x \left(-36 a^2 b^2+3 a^4+40 b^4\right)}{8 a^6}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}",1,"((3*a^4 - 36*a^2*b^2 + 40*b^4)*x)/(8*a^6) - (2*Sqrt[a - b]*b*Sqrt[a + b]*(2*a^2 - 5*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*d) + (b*(11*a^2 - 15*b^2)*Sin[c + d*x])/(3*a^5*d) - ((13*a^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^4*d) + ((3*a^2 - 5*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^3*b*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) - ((a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(a^2*b*d*(b + a*Cos[c + d*x]))","A",8,7,21,0.3333,1,"{3872, 2892, 3049, 3023, 2735, 2659, 208}"
218,1,152,0,0.5709668,"\int \frac{\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Sin[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","-\frac{2 b \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)}{a^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2-6 b^2\right)}{2 a^4}+\frac{3 b \sin (c+d x)}{a^3 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}","-\frac{2 b \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)}{a^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2-6 b^2\right)}{2 a^4}+\frac{3 b \sin (c+d x)}{a^3 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}",1,"((a^2 - 6*b^2)*x)/(2*a^4) - (2*b*(2*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + (3*b*Sin[c + d*x])/(a^3*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(a*d*(b + a*Cos[c + d*x]))","A",8,8,21,0.3810,1,"{3872, 2889, 3048, 3050, 3023, 2735, 2659, 208}"
219,1,203,0,0.4357044,"\int \frac{\csc ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{a b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 a^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)^{5/2} (a+b)^{5/2}}-\frac{2 b^3 \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{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","\frac{a b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 a^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)^{5/2} (a+b)^{5/2}}-\frac{2 b^3 \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{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"(-4*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (a*b^2*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",11,7,21,0.3333,1,"{3872, 2731, 2648, 2664, 12, 2659, 208}"
220,1,343,0,0.5471659,"\int \frac{\csc ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[Csc[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{4 a^2 b \left(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{2 a^2 b^3 \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{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}-\frac{(a-b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}+\frac{(a+b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}","\frac{a^3 b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{4 a^2 b \left(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{2 a^2 b^3 \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{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}-\frac{(a-b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}+\frac{(a+b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}",1,"(-2*a^2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (4*a^2*b*(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) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])^2) - ((a - b)*Sin[c + d*x])/(4*(a + b)^3*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])^2) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])) + ((a + b)*Sin[c + d*x])/(4*(a - b)^3*d*(1 + Cos[c + d*x])) + (a^3*b^2*Sin[c + d*x])/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))","A",15,8,21,0.3810,1,"{3872, 2897, 2650, 2648, 2664, 12, 2659, 208}"
221,1,329,0,0.5021693,"\int \frac{\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]","-\frac{3 \left(a^2-2 b^2\right) \cos ^5(c+d x)}{5 a^5 d}+\frac{b \left(9 a^2-10 b^2\right) \cos ^4(c+d x)}{4 a^6 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \cos ^3(c+d x)}{a^7 d}-\frac{3 b \left(-10 a^2 b^2+3 a^4+7 b^4\right) \cos ^2(c+d x)}{2 a^8 d}-\frac{\left(-18 a^4 b^2+45 a^2 b^4+a^6-28 b^6\right) \cos (c+d x)}{a^9 d}+\frac{3 b^2 \left(a^2-3 b^2\right) \left(a^2-b^2\right)^2}{a^{10} d (a \cos (c+d x)+b)}-\frac{b^3 \left(a^2-b^2\right)^3}{2 a^{10} d (a \cos (c+d x)+b)^2}+\frac{3 b \left(a^2-b^2\right) \left(-9 a^2 b^2+a^4+12 b^4\right) \log (a \cos (c+d x)+b)}{a^{10} d}-\frac{b \cos ^6(c+d x)}{2 a^4 d}+\frac{\cos ^7(c+d x)}{7 a^3 d}","-\frac{3 \left(a^2-2 b^2\right) \cos ^5(c+d x)}{5 a^5 d}+\frac{b \left(9 a^2-10 b^2\right) \cos ^4(c+d x)}{4 a^6 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \cos ^3(c+d x)}{a^7 d}-\frac{3 b \left(-10 a^2 b^2+3 a^4+7 b^4\right) \cos ^2(c+d x)}{2 a^8 d}-\frac{\left(-18 a^4 b^2+45 a^2 b^4+a^6-28 b^6\right) \cos (c+d x)}{a^9 d}+\frac{3 b^2 \left(a^2-3 b^2\right) \left(a^2-b^2\right)^2}{a^{10} d (a \cos (c+d x)+b)}-\frac{b^3 \left(a^2-b^2\right)^3}{2 a^{10} d (a \cos (c+d x)+b)^2}+\frac{3 b \left(a^2-b^2\right) \left(-9 a^2 b^2+a^4+12 b^4\right) \log (a \cos (c+d x)+b)}{a^{10} d}-\frac{b \cos ^6(c+d x)}{2 a^4 d}+\frac{\cos ^7(c+d x)}{7 a^3 d}",1,"-(((a^6 - 18*a^4*b^2 + 45*a^2*b^4 - 28*b^6)*Cos[c + d*x])/(a^9*d)) - (3*b*(3*a^4 - 10*a^2*b^2 + 7*b^4)*Cos[c + d*x]^2)/(2*a^8*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(a^7*d) + (b*(9*a^2 - 10*b^2)*Cos[c + d*x]^4)/(4*a^6*d) - (3*(a^2 - 2*b^2)*Cos[c + d*x]^5)/(5*a^5*d) - (b*Cos[c + d*x]^6)/(2*a^4*d) + Cos[c + d*x]^7/(7*a^3*d) - (b^3*(a^2 - b^2)^3)/(2*a^10*d*(b + a*Cos[c + d*x])^2) + (3*b^2*(a^2 - 3*b^2)*(a^2 - b^2)^2)/(a^10*d*(b + a*Cos[c + d*x])) + (3*b*(a^2 - b^2)*(a^4 - 9*a^2*b^2 + 12*b^4)*Log[b + a*Cos[c + d*x]])/(a^10*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
222,1,239,0,0.3647607,"\int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left(3 a^2-5 b^2\right) \cos ^2(c+d x)}{a^6 d}-\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \cos (c+d x)}{a^7 d}+\frac{b^2 \left(-10 a^2 b^2+3 a^4+7 b^4\right)}{a^8 d (a \cos (c+d x)+b)}-\frac{b^3 \left(a^2-b^2\right)^2}{2 a^8 d (a \cos (c+d x)+b)^2}+\frac{b \left(-20 a^2 b^2+3 a^4+21 b^4\right) \log (a \cos (c+d x)+b)}{a^8 d}+\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d}","\frac{2 \left(a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^5 d}-\frac{b \left(3 a^2-5 b^2\right) \cos ^2(c+d x)}{a^6 d}-\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \cos (c+d x)}{a^7 d}+\frac{b^2 \left(-10 a^2 b^2+3 a^4+7 b^4\right)}{a^8 d (a \cos (c+d x)+b)}-\frac{b^3 \left(a^2-b^2\right)^2}{2 a^8 d (a \cos (c+d x)+b)^2}+\frac{b \left(-20 a^2 b^2+3 a^4+21 b^4\right) \log (a \cos (c+d x)+b)}{a^8 d}+\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d}",1,"-(((a^4 - 12*a^2*b^2 + 15*b^4)*Cos[c + d*x])/(a^7*d)) - (b*(3*a^2 - 5*b^2)*Cos[c + d*x]^2)/(a^6*d) + (2*(a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*a^5*d) + (3*b*Cos[c + d*x]^4)/(4*a^4*d) - Cos[c + d*x]^5/(5*a^3*d) - (b^3*(a^2 - b^2)^2)/(2*a^8*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^4 - 10*a^2*b^2 + 7*b^4))/(a^8*d*(b + a*Cos[c + d*x])) + (b*(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]])/(a^8*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 948}"
223,1,158,0,0.2716816,"\int \frac{\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","-\frac{b^3 \left(a^2-b^2\right)}{2 a^6 d (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-5 b^2\right)}{a^6 d (a \cos (c+d x)+b)}-\frac{\left(a^2-6 b^2\right) \cos (c+d x)}{a^5 d}+\frac{b \left(3 a^2-10 b^2\right) \log (a \cos (c+d x)+b)}{a^6 d}-\frac{3 b \cos ^2(c+d x)}{2 a^4 d}+\frac{\cos ^3(c+d x)}{3 a^3 d}","-\frac{b^3 \left(a^2-b^2\right)}{2 a^6 d (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-5 b^2\right)}{a^6 d (a \cos (c+d x)+b)}-\frac{\left(a^2-6 b^2\right) \cos (c+d x)}{a^5 d}+\frac{b \left(3 a^2-10 b^2\right) \log (a \cos (c+d x)+b)}{a^6 d}-\frac{3 b \cos ^2(c+d x)}{2 a^4 d}+\frac{\cos ^3(c+d x)}{3 a^3 d}",1,"-(((a^2 - 6*b^2)*Cos[c + d*x])/(a^5*d)) - (3*b*Cos[c + d*x]^2)/(2*a^4*d) + Cos[c + d*x]^3/(3*a^3*d) - (b^3*(a^2 - b^2))/(2*a^6*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 - 5*b^2))/(a^6*d*(b + a*Cos[c + d*x])) + (b*(3*a^2 - 10*b^2)*Log[b + a*Cos[c + d*x]])/(a^6*d)","A",5,4,21,0.1905,1,"{3872, 2837, 12, 894}"
224,1,83,0,0.1335588,"\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]/(a + b*Sec[c + d*x])^3,x]","-\frac{b^3}{2 a^4 d (a \cos (c+d x)+b)^2}+\frac{3 b^2}{a^4 d (a \cos (c+d x)+b)}+\frac{3 b \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\cos (c+d x)}{a^3 d}","-\frac{b^3}{2 a^4 d (a \cos (c+d x)+b)^2}+\frac{3 b^2}{a^4 d (a \cos (c+d x)+b)}+\frac{3 b \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\cos (c+d x)}{a^3 d}",1,"-(Cos[c + d*x]/(a^3*d)) - b^3/(2*a^4*d*(b + a*Cos[c + d*x])^2) + (3*b^2)/(a^4*d*(b + a*Cos[c + d*x])) + (3*b*Log[b + a*Cos[c + d*x]])/(a^4*d)","A",5,4,19,0.2105,1,"{3872, 2833, 12, 43}"
225,1,163,0,0.3193504,"\int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]/(a + b*Sec[c + d*x])^3,x]","-\frac{b^3}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-b^2\right)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{b \left(3 a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^3}","-\frac{b^3}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-b^2\right)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{b \left(3 a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^3}",1,"-b^3/(2*a^2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 - b^2))/(a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^3*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)^3*d) + (b*(3*a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)","A",5,4,19,0.2105,1,"{3872, 2837, 12, 1629}"
226,1,229,0,0.5125919,"\int \frac{\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","-\frac{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{b \left(8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","-\frac{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{b \left(8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"-b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",5,4,21,0.1905,1,"{3872, 2721, 1647, 1629}"
227,1,313,0,1.0066743,"\int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{3 a^2 b^2 \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}+\frac{3 a^2 b \left(5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^5}+\frac{\csc ^4(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}+\frac{\csc ^2(c+d x) \left(4 b \left(8 a^2 b^2+3 a^4+b^4\right)-3 a \left(10 a^2 b^2+a^4+5 b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}+\frac{3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac{3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5}","\frac{3 a^2 b^2 \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}+\frac{3 a^2 b \left(5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^5}+\frac{\csc ^4(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}+\frac{\csc ^2(c+d x) \left(4 b \left(8 a^2 b^2+3 a^4+b^4\right)-3 a \left(10 a^2 b^2+a^4+5 b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}+\frac{3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac{3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5}",1,"-(a^2*b^3)/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])^2) + (3*a^2*b^2*(a^2 + b^2))/((a^2 - b^2)^4*d*(b + a*Cos[c + d*x])) + ((4*b*(3*a^4 + 8*a^2*b^2 + b^4) - 3*a*(a^4 + 10*a^2*b^2 + 5*b^4)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^4*d) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^3*d) + (3*a*(a - 3*b)*Log[1 - Cos[c + d*x]])/(16*(a + b)^5*d) - (3*a*(a + 3*b)*Log[1 + Cos[c + d*x]])/(16*(a - b)^5*d) + (3*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^5*d)","A",7,5,21,0.2381,1,"{3872, 2837, 12, 1647, 1629}"
228,1,539,0,2.4356703,"\int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^3,x]","\frac{b \left(-985 a^2 b^2+213 a^4+840 b^4\right) \sin (c+d x)}{30 a^8 d}+\frac{\left(-60 a^2 b^2+9 a^4+56 b^4\right) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac{\left(-110 a^2 b^2+15 a^4+112 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac{\left(-169 a^2 b^2+24 a^4+168 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac{\left(-291 a^2 b^2+45 a^4+280 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac{\left(-244 a^2 b^2+43 a^4+224 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^7 d}-\frac{b \sqrt{a-b} \sqrt{a+b} \left(-47 a^2 b^2+6 a^4+56 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^9 d}+\frac{x \left(-180 a^4 b^2+600 a^2 b^4+5 a^6-448 b^6\right)}{16 a^9}+\frac{4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}","\frac{b \left(-985 a^2 b^2+213 a^4+840 b^4\right) \sin (c+d x)}{30 a^8 d}+\frac{\left(-60 a^2 b^2+9 a^4+56 b^4\right) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac{\left(-110 a^2 b^2+15 a^4+112 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac{\left(-169 a^2 b^2+24 a^4+168 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac{\left(-291 a^2 b^2+45 a^4+280 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac{\left(-244 a^2 b^2+43 a^4+224 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^7 d}-\frac{b \sqrt{a-b} \sqrt{a+b} \left(-47 a^2 b^2+6 a^4+56 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^9 d}+\frac{x \left(-180 a^4 b^2+600 a^2 b^4+5 a^6-448 b^6\right)}{16 a^9}+\frac{4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}",1,"((5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*x)/(16*a^9) - (Sqrt[a - b]*b*Sqrt[a + b]*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^9*d) + (b*(213*a^4 - 985*a^2*b^2 + 840*b^4)*Sin[c + d*x])/(30*a^8*d) - ((43*a^4 - 244*a^2*b^2 + 224*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^7*d) + ((45*a^4 - 291*a^2*b^2 + 280*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(30*a^6*b*d) - ((24*a^4 - 169*a^2*b^2 + 168*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^5*b^2*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(4*b*d*(b + a*Cos[c + d*x])^2) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(10*b^2*d*(b + a*Cos[c + d*x])^2) + ((9*a^4 - 60*a^2*b^2 + 56*b^4)*Cos[c + d*x]^5*Sin[c + d*x])/(60*a^3*b^2*d*(b + a*Cos[c + d*x])^2) + (4*b*Cos[c + d*x]^6*Sin[c + d*x])/(15*a^2*d*(b + a*Cos[c + d*x])^2) - (Cos[c + d*x]^7*Sin[c + d*x])/(6*a*d*(b + a*Cos[c + d*x])^2) + ((15*a^4 - 110*a^2*b^2 + 112*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(20*a^4*b^2*d*(b + a*Cos[c + d*x]))","A",11,8,21,0.3810,1,"{3872, 2896, 3047, 3049, 3023, 2735, 2659, 208}"
229,1,333,0,1.1383867,"\int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","\frac{b \left(13 a^2-30 b^2\right) \sin (c+d x)}{2 a^6 d}+\frac{\left(2 a^2-7 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}-\frac{\left(4 a^2-15 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}+\frac{\left(3 a^2-10 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac{3 \left(7 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^5 d}-\frac{3 b \left(-11 a^2 b^2+2 a^4+10 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d \sqrt{a-b} \sqrt{a+b}}+\frac{3 x \left(-24 a^2 b^2+a^4+40 b^4\right)}{8 a^7}","\frac{b \left(13 a^2-30 b^2\right) \sin (c+d x)}{2 a^6 d}+\frac{\left(2 a^2-7 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}-\frac{\left(4 a^2-15 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}+\frac{\left(3 a^2-10 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac{3 \left(7 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^5 d}-\frac{3 b \left(-11 a^2 b^2+2 a^4+10 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d \sqrt{a-b} \sqrt{a+b}}+\frac{3 x \left(-24 a^2 b^2+a^4+40 b^4\right)}{8 a^7}",1,"(3*(a^4 - 24*a^2*b^2 + 40*b^4)*x)/(8*a^7) - (3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^7*Sqrt[a - b]*Sqrt[a + b]*d) + (b*(13*a^2 - 30*b^2)*Sin[c + d*x])/(2*a^6*d) - (3*(7*a^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^5*d) + ((3*a^2 - 10*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^4*b*d) - ((4*a^2 - 15*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b*d*(b + a*Cos[c + d*x])^2) + ((2*a^2 - 7*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b^2*d*(b + a*Cos[c + d*x]))","A",9,7,21,0.3333,1,"{3872, 2891, 3049, 3023, 2735, 2659, 208}"
230,1,267,0,0.9403943,"\int \frac{\sin ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Sin[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{b \left(11 a^2-12 b^2\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\left(5 a^2-6 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{b \left(-19 a^2 b^2+6 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)^{3/2} (a+b)^{3/2}}+\frac{x \left(a^2-12 b^2\right)}{2 a^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 a d (a \cos (c+d x)+b)^2}","\frac{b \left(11 a^2-12 b^2\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\left(5 a^2-6 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{b \left(-19 a^2 b^2+6 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)^{3/2} (a+b)^{3/2}}+\frac{x \left(a^2-12 b^2\right)}{2 a^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 a d (a \cos (c+d x)+b)^2}",1,"((a^2 - 12*b^2)*x)/(2*a^5) - (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(11*a^2 - 12*b^2)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*a*d*(b + a*Cos[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",9,8,21,0.3810,1,"{3872, 2889, 3048, 3049, 3023, 2735, 2659, 208}"
231,1,376,0,0.6583113,"\int \frac{\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{3 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{2 b^3 \left(3 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 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b^3 \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)}{a d (a-b)^{7/2} (a+b)^{7/2}}-\frac{2 a b \left(3 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{\sin (c+d x)}{2 d (a+b)^3 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^3 (\cos (c+d x)+1)}","\frac{3 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2-b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{2 b^3 \left(3 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 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b^3 \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)}{a d (a-b)^{7/2} (a+b)^{7/2}}-\frac{2 a b \left(3 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{\sin (c+d x)}{2 d (a+b)^3 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^3 (\cos (c+d x)+1)}",1,"(-2*b^3*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(7/2)*(a + b)^(7/2)*d) - (2*a*b*(3*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) - (b^3*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(7/2)*(a + b)^(7/2)*d) - Sin[c + d*x]/(2*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^3*d*(1 + Cos[c + d*x])) - (b^3*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (3*b^4*Sin[c + d*x])/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (b^2*(3*a^2 - b^2)*Sin[c + d*x])/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))","A",16,8,21,0.3810,1,"{3872, 2897, 2648, 2664, 12, 2659, 208, 2754}"
232,1,515,0,0.774694,"\int \frac{\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[Csc[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","\frac{3 a^2 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}+\frac{a^2 b^2 \left(3 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{2 a b^3 \left(3 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)^{9/2} (a+b)^{9/2}}-\frac{a b^3 \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)^{9/2} (a+b)^{9/2}}-\frac{2 a b \left(8 a^2 b^2+3 a^4+b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{9/2} (a+b)^{9/2}}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))}-\frac{(a-2 b) \sin (c+d x)}{4 d (a+b)^4 (1-\cos (c+d x))}+\frac{(a+2 b) \sin (c+d x)}{4 d (a-b)^4 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)^2}","\frac{3 a^2 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}+\frac{a^2 b^2 \left(3 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{2 a b^3 \left(3 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)^{9/2} (a+b)^{9/2}}-\frac{a b^3 \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)^{9/2} (a+b)^{9/2}}-\frac{2 a b \left(8 a^2 b^2+3 a^4+b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{9/2} (a+b)^{9/2}}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))}-\frac{(a-2 b) \sin (c+d x)}{4 d (a+b)^4 (1-\cos (c+d x))}+\frac{(a+2 b) \sin (c+d x)}{4 d (a-b)^4 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)^2}",1,"(-2*a*b^3*(3*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d) - (a*b^3*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d) - (2*a*b*(3*a^4 + 8*a^2*b^2 + b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d) - Sin[c + d*x]/(12*(a + b)^3*d*(1 - Cos[c + d*x])^2) - ((a - 2*b)*Sin[c + d*x])/(4*(a + b)^4*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(12*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^3*d*(1 + Cos[c + d*x])^2) + Sin[c + d*x]/(12*(a - b)^3*d*(1 + Cos[c + d*x])) + ((a + 2*b)*Sin[c + d*x])/(4*(a - b)^4*d*(1 + Cos[c + d*x])) - (a^2*b^3*Sin[c + d*x])/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])^2) + (3*a^2*b^4*Sin[c + d*x])/(2*(a^2 - b^2)^4*d*(b + a*Cos[c + d*x])) + (a^2*b^2*(3*a^2 + b^2)*Sin[c + d*x])/((a^2 - b^2)^4*d*(b + a*Cos[c + d*x]))","A",20,9,21,0.4286,1,"{3872, 2897, 2650, 2648, 2664, 2754, 12, 2659, 208}"
233,1,516,0,1.7026385,"\int \frac{(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x]),x]","-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left(21 b \left(a^2-b^2\right)-a \left(5 a^2-7 b^2\right) \cos (c+d x)\right)}{21 a^4 d}+\frac{2 e^4 \left(-28 a^2 b^2+5 a^4+21 b^4\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^5 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d}","-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left(21 b \left(a^2-b^2\right)-a \left(5 a^2-7 b^2\right) \cos (c+d x)\right)}{21 a^4 d}+\frac{2 e^4 \left(-28 a^2 b^2+5 a^4+21 b^4\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^5 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d}",1,"-((b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d)) - (b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (2*(5*a^4 - 28*a^2*b^2 + 21*b^4)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^5*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(21*b*(a^2 - b^2) - a*(5*a^2 - 7*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(21*a^4*d) + (2*e*(7*b - 5*a*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*a^2*d)","A",15,12,25,0.4800,1,"{3872, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
234,1,430,0,1.1090321,"\int \frac{(e \sin (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x]),x]","\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}-\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}+\frac{2 e^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^3 d \sqrt{\sin (c+d x)}}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{3/2} (5 b-3 a \cos (c+d x))}{15 a^2 d}","\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}-\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}+\frac{2 e^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^3 d \sqrt{\sin (c+d x)}}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{3/2} (5 b-3 a \cos (c+d x))}{15 a^2 d}",1,"(b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) - (b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) - (b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^4*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^4*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*(3*a^2 - 5*b^2)*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^3*d*Sqrt[Sin[c + d*x]]) + (2*e*(5*b - 3*a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^2*d)","A",14,12,25,0.4800,1,"{3872, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
235,1,444,0,1.0442251,"\int \frac{(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x]),x]","-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}+\frac{2 e^2 \left(a^2-3 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a^3 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 e \sqrt{e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}","-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}+\frac{2 e^2 \left(a^2-3 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a^3 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 e \sqrt{e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}",1,"-((b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*d)) - (b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*d) + (2*(a^2 - 3*b^2)*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^3*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^3*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^3*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*e*(3*b - a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^2*d)","A",14,12,25,0.4800,1,"{3872, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
236,1,356,0,0.76152,"\int \frac{\sqrt{e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}-\frac{b \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}","\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}-\frac{b \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}",1,"(b*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b^2*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^2*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]])","A",13,11,25,0.4400,1,"{3872, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
237,1,370,0,0.7808441,"\int \frac{1}{(a+b \sec (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + b*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{e \sin (c+d x)}}","-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{e \sin (c+d x)}}",1,"-((b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(3/4)*d*Sqrt[e])) - (b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]])","A",13,11,25,0.4400,1,"{3872, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
238,1,430,0,1.0368508,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]","\frac{\sqrt{a} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{\sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}","\frac{\sqrt{a} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{\sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}",1,"(Sqrt[a]*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(5/4)*d*e^(3/2)) - (Sqrt[a]*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(5/4)*d*e^(3/2)) + (2*(b - a*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - (b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]])","A",14,12,25,0.4800,1,"{3872, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
239,1,452,0,1.0503696,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","-\frac{a^{3/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}-\frac{a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2}}","-\frac{a^{3/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}-\frac{a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2}}",1,"-((a^(3/2)*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2))) - (a^(3/2)*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) + (2*(b - a*Cos[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) + (a*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (a*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]])","A",14,12,25,0.4800,1,"{3872, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
240,1,511,0,1.3766088,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx","Int[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(7/2)),x]","\frac{a^{5/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}-\frac{a^{5/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}+\frac{2 \left(5 a^2 b-a \left(3 a^2+2 b^2\right) \cos (c+d x)\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}-\frac{2 a \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{5 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2}}","\frac{a^{5/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}-\frac{a^{5/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}+\frac{2 \left(5 a^2 b-a \left(3 a^2+2 b^2\right) \cos (c+d x)\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}-\frac{2 a \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{5 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2}}",1,"(a^(5/2)*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(9/4)*d*e^(7/2)) - (a^(5/2)*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(9/4)*d*e^(7/2)) + (2*(b - a*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(5/2)) + (2*(5*a^2*b - a*(3*a^2 + 2*b^2)*Cos[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Sin[c + d*x]]) - (a^2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(a - Sqrt[a^2 - b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (a^2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(a + Sqrt[a^2 - b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*a*(3*a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Sin[c + d*x]])","A",15,12,25,0.4800,1,"{3872, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
241,1,1070,0,2.7946039,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(9/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b \left(a^2-b^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}-\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{2 b \left(a^2-b^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}+\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{7 b^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left(8 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}+\frac{14 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac{7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac{4 b \left(5 \left(a^2-b^2\right)+3 a b \cos (c+d x)\right) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac{4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac{b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))}","-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b \left(a^2-b^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}-\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{2 b \left(a^2-b^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}+\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{7 b^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left(8 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}+\frac{14 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac{7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac{4 b \left(5 \left(a^2-b^2\right)+3 a b \cos (c+d x)\right) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac{4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac{b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))}",1,"(-7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(13/2)*d) + (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(13/2)*d) + (7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(13/2)*d) - (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(13/2)*d) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (14*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(15*a^2*d*Sqrt[Sin[c + d*x]]) - (7*b^2*(3*a^2 - 5*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]]) - (4*b^2*(8*a^2 - 5*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]]) - (14*e^3*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(45*a^2*d) - (7*b^2*e^3*(5*b - 3*a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^5*d) + (4*b*e^3*(5*(a^2 - b^2) + 3*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(7/2))/(7*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(7/2))/(9*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(7/2))/(a^3*d*(b + a*Cos[c + d*x]))","A",35,16,25,0.6400,1,"{3872, 2912, 2635, 2640, 2639, 2693, 2865, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2695}"
242,1,1101,0,2.9304708,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{5 b^2 \left(a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{10 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{10 \cos (c+d x) \sqrt{e \sin (c+d x)} e^3}{21 a^2 d}-\frac{5 b^2 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}+\frac{4 b \left(3 \left(a^2-b^2\right)+a b \cos (c+d x)\right) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{5/2} e}{7 a^2 d}+\frac{4 b (e \sin (c+d x))^{5/2} e}{5 a^3 d}+\frac{b^2 (e \sin (c+d x))^{5/2} e}{a^3 d (b+a \cos (c+d x))}","-\frac{5 b^2 \left(a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{10 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{10 \cos (c+d x) \sqrt{e \sin (c+d x)} e^3}{21 a^2 d}-\frac{5 b^2 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}+\frac{4 b \left(3 \left(a^2-b^2\right)+a b \cos (c+d x)\right) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{5/2} e}{7 a^2 d}+\frac{4 b (e \sin (c+d x))^{5/2} e}{5 a^3 d}+\frac{b^2 (e \sin (c+d x))^{5/2} e}{a^3 d (b+a \cos (c+d x))}",1,"(5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(11/2)*d) + (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(11/2)*d) + (10*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]]) - (5*b^2*(a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c + d*x]]) - (4*b^2*(4*a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (10*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) - (5*b^2*e^3*(3*b - a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e^3*(3*(a^2 - b^2) + a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(5/2))/(5*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(5/2))/(a^3*d*(b + a*Cos[c + d*x]))","A",35,16,25,0.6400,1,"{3872, 2912, 2635, 2642, 2641, 2693, 2865, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2695}"
243,1,850,0,2.1264813,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2,x]","\frac{3 e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{3 e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{3 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{7 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^4 d \sqrt{\sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{6 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}","\frac{3 e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{3 e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{3 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{7 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^4 d \sqrt{\sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{6 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}",1,"(-3*b^3*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d) + (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (3*b^3*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d) - (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (3*b^4*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^5*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^5*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (6*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]]) - (7*b^2*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^4*d*Sqrt[Sin[c + d*x]]) + (4*b*e*(e*Sin[c + d*x])^(3/2))/(3*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(3/2))/(a^3*d*(b + a*Cos[c + d*x]))","A",32,15,25,0.6000,1,"{3872, 2912, 2635, 2640, 2639, 2693, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2695}"
244,1,882,0,2.1678216,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2,x]","-\frac{e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e \sqrt{e \sin (c+d x)} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{5 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}+\frac{4 e \sqrt{e \sin (c+d x)} b}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{2 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}","-\frac{e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e \sqrt{e \sin (c+d x)} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{5 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}+\frac{4 e \sqrt{e \sin (c+d x)} b}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{2 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}",1,"(b^3*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) + (b^3*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) + (2*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*d*Sqrt[e*Sin[c + d*x]]) - (5*b^2*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^4*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^4*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^4*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^4*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^4*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (4*b*e*Sqrt[e*Sin[c + d*x]])/(a^3*d) - (2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a^2*d) + (b^2*e*Sqrt[e*Sin[c + d*x]])/(a^3*d*(b + a*Cos[c + d*x]))","A",32,15,25,0.6000,1,"{3872, 2912, 2635, 2642, 2641, 2693, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2695}"
245,1,809,0,1.8332617,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","-\frac{e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}+\frac{(e \sin (c+d x))^{3/2} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}-\frac{E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{\sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}-\frac{2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d \sqrt{\sin (c+d x)}}","-\frac{e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}+\frac{(e \sin (c+d x))^{3/2} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}-\frac{E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{\sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}-\frac{2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d \sqrt{\sin (c+d x)}}",1,"(b^3*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) + (2*b*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (b^3*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) - (2*b*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (2*b^2*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^3*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^3*(a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^3*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^3*(a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*d*Sqrt[Sin[c + d*x]]) - (b^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) + (b^2*(e*Sin[c + d*x])^(3/2))/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))","A",27,13,25,0.5200,1,"{3872, 2912, 2640, 2639, 2694, 2867, 2701, 2807, 2805, 329, 298, 205, 208}"
246,1,838,0,1.9240361,"\int \frac{1}{(a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),x]","\frac{3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}+\frac{\sqrt{e \sin (c+d x)} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}+\frac{F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}","\frac{3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}+\frac{\sqrt{e \sin (c+d x)} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}+\frac{F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}",1,"(-3*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqrt[e]) - (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*Sqrt[e]) - (3*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqrt[e]) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*Sqrt[e*Sin[c + d*x]])/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))","A",27,13,25,0.5200,1,"{3872, 2912, 2642, 2641, 2694, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
247,1,1054,0,2.6918178,"\int \frac{1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),x]","-\frac{5 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{5 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{\left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left(5 a b-\left(3 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{a^2 \left(a^2-b^2\right)^2 d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}+\frac{4 (a-b \cos (c+d x)) b}{a^2 \left(a^2-b^2\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \cos (c+d x)}{a^2 d e \sqrt{e \sin (c+d x)}}","-\frac{5 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{5 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{\left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left(5 a b-\left(3 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{a^2 \left(a^2-b^2\right)^2 d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}+\frac{4 (a-b \cos (c+d x)) b}{a^2 \left(a^2-b^2\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \cos (c+d x)}{a^2 d e \sqrt{e \sin (c+d x)}}",1,"(5*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(9/4)*d*e^(3/2)) + (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (5*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(9/4)*d*e^(3/2)) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (2*Cos[c + d*x])/(a^2*d*e*Sqrt[e*Sin[c + d*x]]) + b^2/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (4*b*(a - b*Cos[c + d*x]))/(a^2*(a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*a*b - (3*a^2 + 2*b^2)*Cos[c + d*x]))/(a^2*(a^2 - b^2)^2*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*d*e^2*Sqrt[Sin[c + d*x]]) - (4*b^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]]) - (b^2*(3*a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)^2*d*e^2*Sqrt[Sin[c + d*x]])","A",33,16,25,0.6400,1,"{3872, 2912, 2636, 2640, 2639, 2694, 2866, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2696}"
248,1,1089,0,2.7808929,"\int \frac{1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","\frac{7 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{7 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}-\frac{7 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}-\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}+\frac{\left(5 a^2+2 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{e \sin (c+d x)}}+\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{\left(7 a b-\left(5 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{3 a^2 \left(a^2-b^2\right)^2 d e (e \sin (c+d x))^{3/2}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}+\frac{4 (a-b \cos (c+d x)) b}{3 a^2 \left(a^2-b^2\right) d e (e \sin (c+d x))^{3/2}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}","\frac{7 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{7 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}-\frac{7 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}-\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}+\frac{\left(5 a^2+2 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{e \sin (c+d x)}}+\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{\left(7 a b-\left(5 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{3 a^2 \left(a^2-b^2\right)^2 d e (e \sin (c+d x))^{3/2}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}+\frac{4 (a-b \cos (c+d x)) b}{3 a^2 \left(a^2-b^2\right) d e (e \sin (c+d x))^{3/2}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}",1,"(-7*Sqrt[a]*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)^(11/4)*d*e^(5/2)) - (2*Sqrt[a]*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (7*Sqrt[a]*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)^(11/4)*d*e^(5/2)) - (2*Sqrt[a]*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (2*Cos[c + d*x])/(3*a^2*d*e*(e*Sin[c + d*x])^(3/2)) + b^2/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (4*b*(a - b*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (b^2*(7*a*b - (5*a^2 + 2*b^2)*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + (2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (4*b^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*a^2 + 2*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]])","A",33,16,25,0.6400,1,"{3872, 2912, 2636, 2642, 2641, 2694, 2866, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2696}"
249,1,125,0,0.0316882,"\int \sqrt{a+b \sec (e+f x)} \, dx","Int[Sqrt[a + b*Sec[e + f*x]],x]","-\frac{2 \cot (e+f x) \sqrt{-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt{\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{a-b}{a+b}\right)}{f \sqrt{a+b}}","-\frac{2 \cot (e+f x) \sqrt{-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt{\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{a-b}{a+b}\right)}{f \sqrt{a+b}}",1,"(-2*Cot[e + f*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[e + f*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[e + f*x]))/(a + b*Sec[e + f*x]))]*Sqrt[(b*(1 + Sec[e + f*x]))/(a + b*Sec[e + f*x])]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f)","A",1,1,14,0.07143,1,"{3780}"
250,1,121,0,0.1149543,"\int \csc ^2(e+f x) \sqrt{a+b \sec (e+f x)} \, dx","Int[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]","\frac{\sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \sec (e+f x)}}{f}","\frac{\sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \sec (e+f x)}}{f}",1,"(Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/f","A",2,2,23,0.08696,1,"{3875, 3832}"
251,1,309,0,0.230189,"\int (a+b \sec (e+f x))^{3/2} \, dx","Int[(a + b*Sec[e + f*x])^(3/2),x]","\frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}","\frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}",1,"(-2*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*(2*a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f","A",5,5,14,0.3571,1,"{3781, 3921, 3784, 3832, 4004}"
252,1,228,0,0.2401744,"\int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx","Int[Csc[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2),x]","-\frac{\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}","-\frac{\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}",1,"(-3*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (3*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*(a + b*Sec[e + f*x])^(3/2))/f","A",4,4,23,0.1739,1,"{3875, 3829, 3832, 4004}"
253,1,106,0,0.0217945,"\int \frac{1}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[1/Sqrt[a + b*Sec[e + f*x]],x]","-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}","-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}",1,"(-2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f)","A",1,1,14,0.07143,1,"{3784}"
254,1,255,0,0.3204154,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx","Int[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]","\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x)}{f \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}+\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}","\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x)}{f \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}+\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}",1,"(Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(Sqrt[a + b]*f) - (Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(Sqrt[a + b]*f) - Cot[e + f*x]/(f*Sqrt[a + b*Sec[e + f*x]]) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])","A",6,6,23,0.2609,1,"{3875, 3833, 21, 3829, 3832, 4004}"
255,1,347,0,0.3328687,"\int \frac{1}{(a+b \sec (e+f x))^{3/2}} \, dx","Int[(a + b*Sec[e + f*x])^(-3/2),x]","\frac{2 b^2 \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}+\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}","\frac{2 b^2 \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}+\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}",1,"(2*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*Sqrt[a + b]*f) - (2*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*Sqrt[a + b]*f) - (2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^2*f) + (2*b^2*Tan[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])","A",6,6,14,0.4286,1,"{3785, 4058, 3921, 3784, 3832, 4004}"
256,1,318,0,0.5245301,"\int \frac{\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]","\frac{4 a b^2 \tan (e+f x)}{f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}+\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac{(3 a-b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}+\frac{4 a \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}","\frac{4 a b^2 \tan (e+f x)}{f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}+\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac{(3 a-b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}+\frac{4 a \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}",1,"(4*a*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*f) - ((3*a - b)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*f) - Cot[e + f*x]/(f*(a + b*Sec[e + f*x])^(3/2)) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (4*a*b^2*Tan[e + f*x])/((a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])","A",6,6,23,0.2609,1,"{3875, 3833, 4003, 4005, 3832, 4004}"
257,1,249,0,0.3869781,"\int (a+b \sec (c+d x))^3 (e \sin (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^3*(e*Sin[c + d*x])^m,x]","\frac{3 a^2 b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}","\frac{3 a^2 b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^2*b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (b^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (3*a*b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))","A",9,6,23,0.2609,1,"{3872, 2912, 2643, 2564, 364, 2577}"
258,1,190,0,0.8393719,"\int (a+b \sec (c+d x))^2 (e \sin (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^m,x]","\frac{a^2 \sin (c+d x) \cos (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \tan (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1)}","\frac{a^2 \sin (c+d x) \cos (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \tan (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1)}",1,"(a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]*(e*Sin[c + d*x])^m)/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a*b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))","A",9,8,23,0.3478,1,"{3872, 2911, 2564, 364, 4398, 4401, 2643, 2577}"
259,1,119,0,0.1575834,"\int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]","\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}","\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))","A",5,5,21,0.2381,1,"{3872, 2838, 2564, 364, 2643}"
260,1,232,0,0.2587451,"\int \frac{(e \sin (c+d x))^m}{a+b \sec (c+d x)} \, dx","Int[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x]),x]","\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a d e (m+1) \sqrt{\cos ^2(c+d x)}}-\frac{b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^2 d (1-m)}","\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a d e (m+1) \sqrt{\cos ^2(c+d x)}}-\frac{b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^2 d (1-m)}",1,"-((b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^2*d*(1 - m))) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])","A",4,4,23,0.1739,1,"{3872, 2867, 2643, 2703}"
261,1,405,0,0.4555207,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^2,x]","\frac{b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (2-m) (a \cos (c+d x)+b)}-\frac{2 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^2 d e (m+1) \sqrt{\cos ^2(c+d x)}}","\frac{b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (2-m) (a \cos (c+d x)+b)}-\frac{2 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^2 d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(-2*b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(1 - m)) + (b^2*e*AppellF1[2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(2 - m)*(b + a*Cos[c + d*x])) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a^2*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])","A",6,4,23,0.1739,1,"{3872, 2912, 2643, 2703}"
262,1,580,0,0.5917562,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^3,x]","\frac{3 b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (2-m) (a \cos (c+d x)+b)}-\frac{b^3 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(3-m;\frac{1-m}{2},\frac{1-m}{2};4-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (3-m) (a \cos (c+d x)+b)^2}-\frac{3 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^3 d e (m+1) \sqrt{\cos ^2(c+d x)}}","\frac{3 b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (2-m) (a \cos (c+d x)+b)}-\frac{b^3 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(3-m;\frac{1-m}{2},\frac{1-m}{2};4-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (3-m) (a \cos (c+d x)+b)^2}-\frac{3 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^3 d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(-3*b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(1 - m)) - (b^3*e*AppellF1[3 - m, (1 - m)/2, (1 - m)/2, 4 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(3 - m)*(b + a*Cos[c + d*x])^2) + (3*b^2*e*AppellF1[2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(2 - m)*(b + a*Cos[c + d*x])) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a^3*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])","A",7,4,23,0.1739,1,"{3872, 2912, 2643, 2703}"
263,0,0,0,0.0676679,"\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m,x]","\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","\text{Int}\left((a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
264,0,0,0,0.0620886,"\int \sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(e*Sin[c + d*x])^m,x]","\int \sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m \, dx","\text{Int}\left(\sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m,x\right)",0,"Defer[Int][Sqrt[a + b*Sec[c + d*x]]*(e*Sin[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
265,0,0,0,0.0696136,"\int \frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(e*Sin[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Defer[Int][(e*Sin[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]","A",0,0,0,0,-1,"{}"
266,0,0,0,0.0709144,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Defer[Int][(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
267,0,0,0,0.0439409,"\int (a+b \sec (c+d x))^n (e \sin (c+d x))^m \, dx","Int[(a + b*Sec[c + d*x])^n*(e*Sin[c + d*x])^m,x]","\int (a+b \sec (c+d x))^n (e \sin (c+d x))^m \, dx","\text{Int}\left((e \sin (c+d x))^m (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*(e*Sin[c + d*x])^m, x]","A",0,0,0,0,-1,"{}"
268,1,150,0,0.1255132,"\int (a+b \sec (c+d x))^n \sin ^5(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^5,x]","-\frac{2 b^3 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(4,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^4 d (n+1)}+\frac{b^5 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(6,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^6 d (n+1)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}","-\frac{2 b^3 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(4,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^4 d (n+1)}+\frac{b^5 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(6,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^6 d (n+1)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^2*d*(1 + n)) - (2*b^3*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^4*d*(1 + n)) + (b^5*Hypergeometric2F1[6, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^6*d*(1 + n))","A",6,3,21,0.1429,1,"{3874, 180, 65}"
269,1,121,0,0.1047895,"\int (a+b \sec (c+d x))^n \sin ^3(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^3,x]","\frac{b \left(6 a^2-b^2 \left(n^2-3 n+2\right)\right) (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}+\frac{\cos ^3(c+d x) (2 a-b (2-n) \sec (c+d x)) (a+b \sec (c+d x))^{n+1}}{6 a^2 d}","\frac{b \left(6 a^2-b^2 \left(n^2-3 n+2\right)\right) (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}+\frac{\cos ^3(c+d x) (2 a-b (2-n) \sec (c+d x)) (a+b \sec (c+d x))^{n+1}}{6 a^2 d}",1,"(b*(6*a^2 - b^2*(2 - 3*n + n^2))*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(6*a^4*d*(1 + n)) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(1 + n)*(2*a - b*(2 - n)*Sec[c + d*x]))/(6*a^2*d)","A",3,3,21,0.1429,1,"{3874, 145, 65}"
270,1,48,0,0.0394676,"\int (a+b \sec (c+d x))^n \sin (c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x],x]","\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}","\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^2*d*(1 + n))","A",2,2,19,0.1053,1,"{3874, 65}"
271,1,115,0,0.1189022,"\int \csc (c+d x) (a+b \sec (c+d x))^n \, dx","Int[Csc[c + d*x]*(a + b*Sec[c + d*x])^n,x]","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}",1,"(Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n))","A",6,4,19,0.2105,1,"{3874, 73, 712, 68}"
272,1,231,0,0.1946997,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^n,x]","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}",1,"(Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)^2*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)^2*d*(1 + n))","A",9,4,21,0.1905,1,"{3874, 180, 68, 712}"
273,0,0,0,0.0398236,"\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4,x]","\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx","\text{Int}\left(\sin ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4, x]","A",0,0,0,0,-1,"{}"
274,0,0,0,0.0398864,"\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2,x]","\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx","\text{Int}\left(\sin ^2(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2, x]","A",0,0,0,0,-1,"{}"
275,1,136,0,0.1648578,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^n,x]","\frac{\sqrt{2} b n \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},1-n;\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}}-\frac{\cot (c+d x) (a+b \sec (c+d x))^n}{d}","\frac{\sqrt{2} b n \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},1-n;\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}}-\frac{\cot (c+d x) (a+b \sec (c+d x))^n}{d}",1,"-((Cot[c + d*x]*(a + b*Sec[c + d*x])^n)/d) + (Sqrt[2]*b*n*AppellF1[1/2, 1/2, 1 - n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/((a + b)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^n)","A",4,4,21,0.1905,1,"{3875, 3834, 139, 138}"
276,0,0,0,0.0400988,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx","Int[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]","\int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx","-\frac{\cot ^3(c+d x) (\sec (c+d x)+1)^{3/2} (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(-\frac{3}{2};\frac{5}{2},-n;-\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{6 \sqrt{2} d}-\frac{3 \cot (c+d x) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(-\frac{1}{2};\frac{5}{2},-n;\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} d \sqrt{\sec (c+d x)+1}}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} d \sqrt{\sec (c+d x)+1}}",1,"Defer[Int][Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n, x]","F",0,0,0,0,-1,"{}"
277,0,0,0,0.0387082,"\int (a+b \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","Int[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^(3/2),x]","\int (a+b \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","\text{Int}\left(\sin ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Sin[c + d*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
278,0,0,0,0.0401989,"\int (a+b \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]],x]","\int (a+b \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","\text{Int}\left(\sqrt{\sin (c+d x)} (a+b \sec (c+d x))^n,x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]], x]","A",0,0,0,0,-1,"{}"
279,0,0,0,0.0404432,"\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","Int[(a + b*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]],x]","\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]], x]","A",0,0,0,0,-1,"{}"
280,0,0,0,0.0408955,"\int \frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","Int[(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2),x]","\int \frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)},x\right)",0,"Defer[Int][(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
281,1,190,0,0.165207,"\int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx","Int[(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]","-\frac{2 a e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}","-\frac{2 a e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}",1,"(-2*a*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) - (2*a*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) + (a*e^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*e^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d)","A",11,11,23,0.4783,1,"{3878, 3872, 2838, 2564, 325, 329, 212, 206, 203, 2636, 2641}"
282,1,169,0,0.1622299,"\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx","Int[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]","-\frac{2 a e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{2 a e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}","-\frac{2 a e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{2 a e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(-2*a*e*Sqrt[e*Csc[c + d*x]])/d - (2*a*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (a*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (2*a*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d","A",11,11,23,0.4783,1,"{3878, 3872, 2838, 2564, 325, 329, 298, 203, 206, 2636, 2639}"
283,1,121,0,0.137354,"\int \sqrt{e \csc (c+d x)} (a+a \sec (c+d x)) \, dx","Int[Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x]),x]","\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}","\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(a*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d","A",9,9,23,0.3913,1,"{3878, 3872, 2838, 2564, 329, 212, 206, 203, 2641}"
284,1,122,0,0.1480307,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \csc (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])/Sqrt[e*Csc[c + d*x]],x]","-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"-((a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",9,9,23,0.3913,1,"{3878, 3872, 2838, 2564, 329, 298, 203, 206, 2639}"
285,1,182,0,0.1722814,"\int \frac{a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(3/2),x]","-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-2*a)/(d*e*Sqrt[e*Csc[c + d*x]]) - (2*a*Cos[c + d*x])/(3*d*e*Sqrt[e*Csc[c + d*x]]) + (a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",11,11,23,0.4783,1,"{3878, 3872, 2838, 2564, 321, 329, 212, 206, 203, 2635, 2641}"
286,1,197,0,0.1721739,"\int \frac{a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(5/2),x]","-\frac{2 a \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{6 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{2 a \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{6 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"-((a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (6*a*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (2*a*Sin[c + d*x])/(3*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*e^2*Sqrt[e*Csc[c + d*x]])","A",11,11,23,0.4783,1,"{3878, 3872, 2838, 2564, 321, 329, 298, 203, 206, 2635, 2639}"
287,1,270,0,0.3339413,"\int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx","Int[(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{5 a^2 e^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{7 a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}","-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{5 a^2 e^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{7 a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}",1,"(-2*a^2*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) - (4*a^2*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) - (2*a^2*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]]*Sec[c + d*x])/(3*d) + (2*a^2*e^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*e^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (7*a^2*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d) + (5*a^2*e^2*Sqrt[e*Csc[c + d*x]]*Tan[c + d*x])/(3*d)","A",15,13,25,0.5200,1,"{3878, 3872, 2873, 2636, 2641, 2564, 325, 329, 212, 206, 203, 2570, 2571}"
288,1,240,0,0.3308452,"\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx","Int[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sec (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{5 a^2 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}","-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sec (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{5 a^2 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(-4*a^2*e*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Sqrt[e*Csc[c + d*x]]*Sec[c + d*x])/d - (2*a^2*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (5*a^2*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d + (3*a^2*e*Sqrt[e*Csc[c + d*x]]*Sin[c + d*x]*Tan[c + d*x])/d","A",15,13,25,0.5200,1,"{3878, 3872, 2873, 2636, 2639, 2564, 325, 329, 298, 203, 206, 2570, 2571}"
289,1,154,0,0.2627066,"\int \sqrt{e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx","Int[Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}","\frac{a^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(2*a^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (3*a^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d + (a^2*Sqrt[e*Csc[c + d*x]]*Tan[c + d*x])/d","A",12,10,25,0.4000,1,"{3878, 3872, 2873, 2641, 2564, 329, 212, 206, 203, 2571}"
290,1,153,0,0.2717038,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \csc (c+d x)}} \, dx","Int[(a + a*Sec[c + d*x])^2/Sqrt[e*Csc[c + d*x]],x]","\frac{a^2 \tan (c+d x)}{d \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{a^2 \tan (c+d x)}{d \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a^2*EllipticE[(c - Pi/2 + d*x)/2, 2])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a^2*Tan[c + d*x])/(d*Sqrt[e*Csc[c + d*x]])","A",12,10,25,0.4000,1,"{3878, 3872, 2873, 2639, 2564, 329, 298, 203, 206, 2571}"
291,1,222,0,0.3125726,"\int \frac{(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]","-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-4*a^2)/(d*e*Sqrt[e*Csc[c + d*x]]) - (2*a^2*Cos[c + d*x])/(3*d*e*Sqrt[e*Csc[c + d*x]]) + (a^2*Sec[c + d*x])/(d*e*Sqrt[e*Csc[c + d*x]]) + (2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",14,12,25,0.4800,1,"{3878, 3872, 2873, 2635, 2641, 2564, 321, 329, 212, 206, 203, 2566}"
292,1,236,0,0.3196959,"\int \frac{(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx","Int[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(5/2),x]","-\frac{4 a^2 \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}+\frac{a^2 \tan (c+d x)}{d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{9 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{4 a^2 \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}+\frac{a^2 \tan (c+d x)}{d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{9 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (9*a^2*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (4*a^2*Sin[c + d*x])/(3*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]) + (a^2*Tan[c + d*x])/(d*e^2*Sqrt[e*Csc[c + d*x]])","A",14,12,25,0.4800,1,"{3878, 3872, 2873, 2635, 2639, 2564, 321, 329, 298, 203, 206, 2566}"
293,1,155,0,0.2235774,"\int \frac{(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","-\frac{2 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}+\frac{2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a d}","-\frac{2 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}+\frac{2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a d}",1,"(-4*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a*d) + (2*e^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a*d) - (2*e^2*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a*d) + (4*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a*d)","A",8,8,25,0.3200,1,"{3878, 3872, 2839, 2564, 30, 2567, 2636, 2641}"
294,1,145,0,0.2293571,"\int \frac{(e \csc (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Int[(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{2 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}+\frac{2 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{5 a d}","-\frac{2 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}+\frac{2 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{5 a d}",1,"(-4*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/(5*a*d) + (2*e*Cot[c + d*x]*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(5*a*d) - (2*e*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(5*a*d) - (4*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(5*a*d)","A",8,8,25,0.3200,1,"{3878, 3872, 2839, 2564, 30, 2567, 2636, 2639}"
295,1,105,0,0.2030625,"\int \frac{\sqrt{e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx","Int[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 a d}","-\frac{2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 a d}",1,"(2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) - (2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) + (4*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a*d)","A",7,7,25,0.2800,1,"{3878, 3872, 2839, 2564, 30, 2567, 2641}"
296,1,99,0,0.2109423,"\int \frac{1}{\sqrt{e \csc (c+d x)} (a+a \sec (c+d x))} \, dx","Int[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{2 \csc (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{2 \cot (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{2 \csc (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{2 \cot (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(2*Cot[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) - (2*Csc[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) + (4*EllipticE[(c - Pi/2 + d*x)/2, 2])/(a*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",7,7,25,0.2800,1,"{3878, 3872, 2839, 2564, 30, 2567, 2639}"
297,1,106,0,0.227105,"\int \frac{1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{2}{a d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{3 a d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{2}{a d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{3 a d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"2/(a*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x])/(3*a*d*e*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*a*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",7,7,25,0.2800,1,"{3878, 3872, 2839, 2564, 30, 2569, 2641}"
298,1,120,0,0.222214,"\int \frac{1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-4*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*a*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*Sin[c + d*x])/(3*a*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*e^2*Sqrt[e*Csc[c + d*x]])","A",7,7,25,0.2800,1,"{3878, 3872, 2839, 2564, 30, 2569, 2639}"
299,1,149,0,0.253985,"\int \frac{1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx","Int[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]","\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-2*Cos[c + d*x])/(21*a*d*e^3*Sqrt[e*Csc[c + d*x]]) + (2*Cos[c + d*x]^3)/(7*a*d*e^3*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(c - Pi/2 + d*x)/2, 2])/(21*a*d*e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*Sin[c + d*x]^2)/(5*a*d*e^3*Sqrt[e*Csc[c + d*x]])","A",8,8,25,0.3200,1,"{3878, 3872, 2839, 2564, 30, 2568, 2569, 2641}"
300,1,268,0,0.5049966,"\int \frac{(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{4 e^2 \csc ^5(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{4 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 e^2 \cot ^3(c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{2 e^2 \cot (c+d x) \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}+\frac{16 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{77 a^2 d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{231 a^2 d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{231 a^2 d}","\frac{4 e^2 \csc ^5(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{4 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 e^2 \cot ^3(c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{2 e^2 \cot (c+d x) \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}+\frac{16 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{77 a^2 d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{231 a^2 d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{231 a^2 d}",1,"(-4*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(231*a^2*d) + (16*e^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(77*a^2*d) - (2*e^2*Cot[c + d*x]^3*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) - (4*e^2*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (2*e^2*Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) + (4*e^2*Csc[c + d*x]^5*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) + (4*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*a^2*d)","A",16,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2636, 2641, 2564, 14}"
301,1,250,0,0.4926833,"\int \frac{(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Int[(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{4 e \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{4 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{15 a^2 d}-\frac{2 e \cot ^3(c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{2 e \cot (c+d x) \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}+\frac{16 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{45 a^2 d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{15 a^2 d}","\frac{4 e \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{4 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{15 a^2 d}-\frac{2 e \cot ^3(c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{2 e \cot (c+d x) \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}+\frac{16 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{45 a^2 d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{15 a^2 d}",1,"(-4*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/(15*a^2*d) + (16*e*Cot[c + d*x]*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(45*a^2*d) - (2*e*Cot[c + d*x]^3*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) - (4*e*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(5*a^2*d) - (2*e*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) + (4*e*Csc[c + d*x]^4*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) - (4*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(15*a^2*d)","A",16,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2636, 2639, 2564, 14}"
302,1,201,0,0.4481387,"\int \frac{\sqrt{e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Int[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a^2 d}","\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a^2 d}",1,"(16*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a^2*d) - (2*Cot[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (4*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a^2*d) - (2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (20*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d)","A",14,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2636, 2641, 2564, 14}"
303,1,199,0,0.4686369,"\int \frac{1}{\sqrt{e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\frac{4 \csc ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{4 \csc (c+d x)}{a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{16 \cot (c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{4 \csc ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{4 \csc (c+d x)}{a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{16 \cot (c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(16*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (4*Csc[c + d*x])/(a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]*Csc[c + d*x]^2)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (28*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*a^2*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",14,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2636, 2639, 2564, 14}"
304,1,213,0,0.4751177,"\int \frac{1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{4 \csc ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}+\frac{4}{a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 \cos (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{4 \csc ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}+\frac{4}{a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 \cos (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"4/(a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (4*Cos[c + d*x])/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x]*Cot[c + d*x]^2)/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x]^2)/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(c - Pi/2 + d*x)/2, 2])/(a^2*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])","A",13,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2641, 2564, 14, 2569}"
305,1,215,0,0.4712646,"\int \frac{1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{4 \csc (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}+\frac{4 \sin (c+d x)}{3 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cos ^2(c+d x) \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{12 \sin (c+d x) \cos (c+d x)}{5 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{44 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","\frac{4 \csc (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}+\frac{4 \sin (c+d x)}{3 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cos ^2(c+d x) \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{12 \sin (c+d x) \cos (c+d x)}{5 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{44 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-2*Cot[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x]^2*Cot[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) - (44*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*a^2*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (4*Sin[c + d*x])/(3*a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) - (12*Cos[c + d*x]*Sin[c + d*x])/(5*a^2*d*e^2*Sqrt[e*Csc[c + d*x]])","A",13,9,25,0.3600,1,"{3878, 3872, 2875, 2873, 2567, 2639, 2564, 14, 2569}"
306,1,172,0,0.4631645,"\int \frac{1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx","Int[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{4}{a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{26 \cos (c+d x)}{21 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{52 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{4}{a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{26 \cos (c+d x)}{21 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{52 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"-4/(a^2*d*e^3*Sqrt[e*Csc[c + d*x]]) + (26*Cos[c + d*x])/(21*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]) + (2*Cos[c + d*x]^3)/(7*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]) + (52*EllipticF[(c - Pi/2 + d*x)/2, 2])/(21*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (4*Sin[c + d*x]^2)/(5*a^2*d*e^3*Sqrt[e*Csc[c + d*x]])","A",13,8,25,0.3200,1,"{3878, 3872, 2875, 2873, 2569, 2641, 2564, 14}"