1,1,115,0,0.0716649,"\int (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}",1,"(-23*a*Log[1 - Sin[c + d*x]])/(16*d) + (7*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 88}"
2,1,71,0,0.048039,"\int (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \sin (c+d x)}{d}+\frac{5 a \log (1-\sin (c+d x))}{4 d}-\frac{a \log (\sin (c+d x)+1)}{4 d}","\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \sin (c+d x)}{d}+\frac{5 a \log (1-\sin (c+d x))}{4 d}-\frac{a \log (\sin (c+d x)+1)}{4 d}",1,"(5*a*Log[1 - Sin[c + d*x]])/(4*d) - (a*Log[1 + Sin[c + d*x]])/(4*d) + (a*Sin[c + d*x])/d + a^2/(2*d*(a - a*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 88}"
3,1,30,0,0.0212762,"\int (a+a \sin (c+d x)) \tan (c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x],x]","-\frac{a \sin (c+d x)}{d}-\frac{a \log (1-\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \log (1-\sin (c+d x))}{d}",1,"-((a*Log[1 - Sin[c + d*x]])/d) - (a*Sin[c + d*x])/d","A",3,2,17,0.1176,1,"{2707, 43}"
4,1,24,0,0.0202604,"\int \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d","A",3,2,17,0.1176,1,"{2707, 43}"
5,1,54,0,0.037076,"\int \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 75}"
6,1,81,0,0.048033,"\int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(2*a*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 88}"
7,1,115,0,0.0594132,"\int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}",1,"(-3*a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 88}"
8,1,101,0,0.0922464,"\int (a+a \sin (c+d x)) \tan ^6(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^6,x]","\frac{a \cos (c+d x)}{d}+\frac{a \tan ^5(c+d x)}{5 d}-\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^5(c+d x)}{5 d}-\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec (c+d x)}{d}-a x","\frac{a \cos (c+d x)}{d}+\frac{a \tan ^5(c+d x)}{5 d}-\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^5(c+d x)}{5 d}-\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec (c+d x)}{d}-a x",1,"-(a*x) + (a*Cos[c + d*x])/d + (3*a*Sec[c + d*x])/d - (a*Sec[c + d*x]^3)/d + (a*Sec[c + d*x]^5)/(5*d) + (a*Tan[c + d*x])/d - (a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)","A",9,5,19,0.2632,1,"{2710, 3473, 8, 2590, 270}"
9,1,72,0,0.0743084,"\int (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}+a x","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}+a x",1,"a*x - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",8,5,19,0.2632,1,"{2710, 3473, 8, 2590, 270}"
10,1,39,0,0.1041278,"\int (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x",1,"-(a*x) + (a*Cos[c + d*x])/d + (a*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",5,5,19,0.2632,1,"{2708, 2746, 12, 2735, 2648}"
11,1,41,0,0.051557,"\int \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos (c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-a x","\frac{a \cos (c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-a x",1,"-(a*x) - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d - (a*Cot[c + d*x])/d","A",7,6,19,0.3158,1,"{2710, 2592, 321, 206, 3473, 8}"
12,1,82,0,0.0797018,"\int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x",1,"a*x + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)","A",9,7,19,0.3684,1,"{2710, 2592, 288, 321, 206, 3473, 8}"
13,1,122,0,0.0962739,"\int \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x","\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x",1,"-(a*x) - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)","A",11,7,19,0.3684,1,"{2710, 2592, 288, 321, 206, 3473, 8}"
14,1,119,0,0.0818173,"\int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-31*a^2*Log[1 - Sin[c + d*x]])/(8*d) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (9*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
15,1,72,0,0.0621593,"\int (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^3}{d (a-a \sin (c+d x))}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{3 a^2 \log (1-\sin (c+d x))}{d}","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^3}{d (a-a \sin (c+d x))}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{3 a^2 \log (1-\sin (c+d x))}{d}",1,"(3*a^2*Log[1 - Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d) + a^3/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 43}"
16,1,52,0,0.0375399,"\int (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{2 a^2 \log (1-\sin (c+d x))}{d}","-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{2 a^2 \log (1-\sin (c+d x))}{d}",1,"(-2*a^2*Log[1 - Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{2707, 77}"
17,1,30,0,0.0392531,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d}","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d}",1,"-(Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4)/(2*a^2*d)","A",2,2,21,0.09524,1,"{2707, 74}"
18,1,132,0,0.0754306,"\int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{a^2 \csc ^4(c+d x)}{2 d}+\frac{2 a^2 \csc ^3(c+d x)}{d}-\frac{6 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}","-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{a^2 \csc ^4(c+d x)}{2 d}+\frac{2 a^2 \csc ^3(c+d x)}{d}-\frac{6 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}",1,"(-6*a^2*Csc[c + d*x])/d + (2*a^2*Csc[c + d*x]^3)/d + (a^2*Csc[c + d*x]^4)/(2*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (2*a^2*Log[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{2707, 88}"
19,1,149,0,0.1651388,"\int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^6,x]","\frac{2 a^2 \cos (c+d x)}{d}+\frac{9 a^2 \tan ^5(c+d x)}{10 d}-\frac{3 a^2 \tan ^3(c+d x)}{2 d}+\frac{9 a^2 \tan (c+d x)}{2 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}-\frac{2 a^2 \sec ^3(c+d x)}{d}+\frac{6 a^2 \sec (c+d x)}{d}-\frac{a^2 \sin ^2(c+d x) \tan ^5(c+d x)}{2 d}-\frac{9 a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}+\frac{9 a^2 \tan ^5(c+d x)}{10 d}-\frac{3 a^2 \tan ^3(c+d x)}{2 d}+\frac{9 a^2 \tan (c+d x)}{2 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}-\frac{2 a^2 \sec ^3(c+d x)}{d}+\frac{6 a^2 \sec (c+d x)}{d}-\frac{a^2 \sin ^2(c+d x) \tan ^5(c+d x)}{2 d}-\frac{9 a^2 x}{2}",1,"(-9*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (6*a^2*Sec[c + d*x])/d - (2*a^2*Sec[c + d*x]^3)/d + (2*a^2*Sec[c + d*x]^5)/(5*d) + (9*a^2*Tan[c + d*x])/(2*d) - (3*a^2*Tan[c + d*x]^3)/(2*d) + (9*a^2*Tan[c + d*x]^5)/(10*d) - (a^2*Sin[c + d*x]^2*Tan[c + d*x]^5)/(2*d)","A",14,9,21,0.4286,1,"{2710, 3473, 8, 2590, 270, 2591, 288, 302, 203}"
20,1,120,0,0.2031429,"\int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{16 a^2 \cos (c+d x)}{3 d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{8 a^2 \sin ^2(c+d x) \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^2 x}{2}","-\frac{16 a^2 \cos (c+d x)}{3 d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{8 a^2 \sin ^2(c+d x) \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^2 x}{2}",1,"(7*a^2*x)/2 - (16*a^2*Cos[c + d*x])/(3*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (8*a^2*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^2)","A",4,4,21,0.1905,1,"{2708, 2765, 2977, 2734}"
21,1,71,0,0.0889338,"\int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2}",1,"(-5*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,5,21,0.2381,1,"{2709, 2648, 2638, 2635, 8}"
22,1,45,0,0.0136893,"\int (a+a \sin (c+d x))^2 \, dx","Int[(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}",1,"(3*a^2*x)/2 - (2*a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",1,1,12,0.08333,1,"{2644}"
23,1,74,0,0.1024711,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 x}{2}",1,"-(a^2*x)/2 - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,21,0.2857,1,"{2709, 3770, 3767, 8, 2638, 2635}"
24,1,98,0,0.1614051,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2}","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2}",1,"-(a^2*x)/2 + (3*a^2*ArcTanh[Cos[c + d*x]])/d - (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",12,7,21,0.3333,1,"{2709, 3770, 3767, 8, 3768, 2638, 2635}"
25,1,160,0,0.1090784,"\int (a+a \sin (c+d x))^3 \tan ^7(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^7,x]","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^6}{6 d (a-a \sin (c+d x))^3}-\frac{13 a^5}{8 d (a-a \sin (c+d x))^2}+\frac{71 a^4}{8 d (a-a \sin (c+d x))}+\frac{7 a^3 \sin (c+d x)}{d}+\frac{209 a^3 \log (1-\sin (c+d x))}{16 d}-\frac{a^3 \log (\sin (c+d x)+1)}{16 d}","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^6}{6 d (a-a \sin (c+d x))^3}-\frac{13 a^5}{8 d (a-a \sin (c+d x))^2}+\frac{71 a^4}{8 d (a-a \sin (c+d x))}+\frac{7 a^3 \sin (c+d x)}{d}+\frac{209 a^3 \log (1-\sin (c+d x))}{16 d}-\frac{a^3 \log (\sin (c+d x)+1)}{16 d}",1,"(209*a^3*Log[1 - Sin[c + d*x]])/(16*d) - (a^3*Log[1 + Sin[c + d*x]])/(16*d) + (7*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + a^6/(6*d*(a - a*Sin[c + d*x])^3) - (13*a^5)/(8*d*(a - a*Sin[c + d*x])^2) + (71*a^4)/(8*d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
26,1,91,0,0.0707841,"\int (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{5 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \log (1-\sin (c+d x))}{d}","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{5 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \log (1-\sin (c+d x))}{d}",1,"(7*a^3*Log[1 - Sin[c + d*x]])/d + (5*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + (2*a^4)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 77}"
27,1,70,0,0.0449772,"\int (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{4 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d}","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{4 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d}",1,"(-4*a^3*Log[1 - Sin[c + d*x]])/d - (4*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2707, 77}"
28,1,98,0,0.0648456,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{2 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{2 a^3 \log (\sin (c+d x))}{d}","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{2 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{2 a^3 \log (\sin (c+d x))}{d}",1,"(-3*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) + (2*a^3*Log[Sin[c + d*x]])/d - (2*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{2707, 75}"
29,1,180,0,0.3566636,"\int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^6,x]","-\frac{136 a^3 \cos ^3(c+d x)}{15 d}+\frac{136 a^3 \cos (c+d x)}{5 d}+\frac{23 a^6 \sin ^3(c+d x) \cos (c+d x)}{3 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{a^6 \sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac{13 a^5 \sin ^4(c+d x) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{23 a^3 x}{2}","-\frac{136 a^3 \cos ^3(c+d x)}{15 d}+\frac{136 a^3 \cos (c+d x)}{5 d}+\frac{23 a^6 \sin ^3(c+d x) \cos (c+d x)}{3 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{a^6 \sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac{13 a^5 \sin ^4(c+d x) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{23 a^3 x}{2}",1,"(-23*a^3*x)/2 + (136*a^3*Cos[c + d*x])/(5*d) - (136*a^3*Cos[c + d*x]^3)/(15*d) + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^6*Cos[c + d*x]*Sin[c + d*x]^5)/(5*d*(a - a*Sin[c + d*x])^3) - (13*a^5*Cos[c + d*x]*Sin[c + d*x]^4)/(15*d*(a - a*Sin[c + d*x])^2) + (23*a^6*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a^3 - a^3*Sin[c + d*x]))","A",9,7,21,0.3333,1,"{2708, 2765, 2977, 2748, 2635, 8, 2633}"
30,1,119,0,0.1941831,"\int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{6 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^3 x}{2}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{6 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^3 x}{2}",1,"(17*a^3*x)/2 - (6*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (25*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",10,7,21,0.3333,1,"{2709, 2650, 2648, 2638, 2635, 8, 2633}"
31,1,89,0,0.1251109,"\int (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 x}{2}","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 x}{2}",1,"(-11*a^3*x)/2 + (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,21,0.2857,1,"{2709, 2648, 2638, 2635, 8, 2633}"
32,1,63,0,0.0544642,"\int (a+a \sin (c+d x))^3 \, dx","Int[(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{4 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{4 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}",1,"(5*a^3*x)/2 - (4*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",7,5,12,0.4167,1,"{2645, 2638, 2635, 8, 2633}"
33,1,92,0,0.1368569,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2}","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2}",1,"(a^3*x)/2 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",10,7,21,0.3333,1,"{2709, 3770, 3767, 8, 2638, 2635, 2633}"
34,1,129,0,0.0947062,"\int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^5,x]","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d}","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{9 a^4 \sin ^2(c+d x)}{2 d}+\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d x))}-\frac{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d}",1,"(-25*a^4*Log[1 - Sin[c + d*x]])/d - (16*a^4*Sin[c + d*x])/d - (9*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d) + a^6/(d*(a - a*Sin[c + d*x])^2) - (11*a^5)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 77}"
35,1,107,0,0.0786493,"\int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{4 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^5}{d (a-a \sin (c+d x))}+\frac{12 a^4 \sin (c+d x)}{d}+\frac{16 a^4 \log (1-\sin (c+d x))}{d}","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{4 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^5}{d (a-a \sin (c+d x))}+\frac{12 a^4 \sin (c+d x)}{d}+\frac{16 a^4 \log (1-\sin (c+d x))}{d}",1,"(16*a^4*Log[1 - Sin[c + d*x]])/d + (12*a^4*Sin[c + d*x])/d + (4*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d) + (4*a^5)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
36,1,88,0,0.0526452,"\int (a+a \sin (c+d x))^4 \tan (c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x],x]","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{7 a^4 \sin ^2(c+d x)}{2 d}-\frac{8 a^4 \sin (c+d x)}{d}-\frac{8 a^4 \log (1-\sin (c+d x))}{d}","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{7 a^4 \sin ^2(c+d x)}{2 d}-\frac{8 a^4 \sin (c+d x)}{d}-\frac{8 a^4 \log (1-\sin (c+d x))}{d}",1,"(-8*a^4*Log[1 - Sin[c + d*x]])/d - (8*a^4*Sin[c + d*x])/d - (7*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d)","A",3,2,19,0.1053,1,"{2707, 77}"
37,1,102,0,0.0667044,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{5 a^4 \sin ^2(c+d x)}{2 d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{5 a^4 \log (\sin (c+d x))}{d}","-\frac{a^4 \sin ^4(c+d x)}{4 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{5 a^4 \sin ^2(c+d x)}{2 d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{5 a^4 \log (\sin (c+d x))}{d}",1,"(-4*a^4*Csc[c + d*x])/d - (a^4*Csc[c + d*x]^2)/(2*d) + (5*a^4*Log[Sin[c + d*x]])/d - (5*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d)","A",3,2,21,0.09524,1,"{2707, 75}"
38,1,143,0,0.1986853,"\int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{16 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{163 a^4 x}{8}","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{16 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{163 a^4 x}{8}",1,"(163*a^4*x)/8 - (16*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (56*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",13,7,21,0.3333,1,"{2709, 2650, 2648, 2638, 2635, 8, 2633}"
39,1,113,0,0.1609164,"\int (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^2,x]","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{12 a^4 \cos (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{31 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{8 a^4 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{95 a^4 x}{8}","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{12 a^4 \cos (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{31 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{8 a^4 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{95 a^4 x}{8}",1,"(-95*a^4*x)/8 + (12*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) + (8*a^4*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (31*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",11,6,21,0.2857,1,"{2709, 2648, 2638, 2635, 8, 2633}"
40,1,87,0,0.0817973,"\int (a+a \sin (c+d x))^4 \, dx","Int[(a + a*Sin[c + d*x])^4,x]","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{8 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{8 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}",1,"(35*a^4*x)/8 - (8*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",10,5,12,0.4167,1,"{2645, 2638, 2635, 8, 2633}"
41,1,116,0,0.1590222,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{23 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{17 a^4 x}{8}","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{23 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{17 a^4 x}{8}",1,"(17*a^4*x)/8 - (4*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x])/d + (23*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",12,6,21,0.2857,1,"{2709, 3770, 3767, 8, 2635, 2633}"
42,1,140,0,0.2249428,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8}","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8}",1,"(-61*a^4*x)/8 + (2*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",17,8,21,0.3810,1,"{2709, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
43,1,198,0,0.4277491,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^4,x]","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{97 a^4 x}{8}","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}-\frac{5 a^4 \cot ^3(c+d x)}{3 d}+\frac{10 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{97 a^4 x}{8}",1,"(97*a^4*x)/8 + (5*a^4*ArcTanh[Cos[c + d*x]])/(2*d) - (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) + (10*a^4*Cot[c + d*x])/d - (5*a^4*Cot[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x]^5)/(5*d) + (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/d + (15*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",21,8,21,0.3810,1,"{2709, 3767, 8, 3768, 3770, 2638, 2635, 2633}"
44,1,130,0,0.1610364,"\int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]","\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d}","\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(-35*ArcTanh[Sin[c + d*x]])/(128*a*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d*x]^8/(8*a*d)","A",8,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
45,1,106,0,0.1362796,"\int \frac{\tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + a*Sin[c + d*x]),x]","\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) \sec (c+d x)}{6 a d}+\frac{5 \tan ^3(c+d x) \sec (c+d x)}{24 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{16 a d}","\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) \sec (c+d x)}{6 a d}+\frac{5 \tan ^3(c+d x) \sec (c+d x)}{24 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{16 a d}",1,"(5*ArcTanh[Sin[c + d*x]])/(16*a*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(16*a*d) + (5*Sec[c + d*x]*Tan[c + d*x]^3)/(24*a*d) - (Sec[c + d*x]*Tan[c + d*x]^5)/(6*a*d) + Tan[c + d*x]^6/(6*a*d)","A",7,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
46,1,82,0,0.1156614,"\int \frac{\tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + a*Sin[c + d*x]),x]","\frac{\tan ^4(c+d x)}{4 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) \sec (c+d x)}{4 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 a d}","\frac{\tan ^4(c+d x)}{4 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{\tan ^3(c+d x) \sec (c+d x)}{4 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(-3*ArcTanh[Sin[c + d*x]])/(8*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) - (Sec[c + d*x]*Tan[c + d*x]^3)/(4*a*d) + Tan[c + d*x]^4/(4*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
47,1,58,0,0.067309,"\int \frac{\tan (c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\sec ^2(c+d x)}{2 a d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{2 a d}","\frac{1}{2 d (a \sin (c+d x)+a)}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}",1,"ArcTanh[Sin[c + d*x]]/(2*a*d) + Sec[c + d*x]^2/(2*a*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",5,5,19,0.2632,1,"{2706, 2606, 30, 2611, 3770}"
48,1,32,0,0.0390786,"\int \frac{\cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{2707, 36, 29, 31}"
49,1,32,0,0.0677649,"\int \frac{\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d)","A",5,4,21,0.1905,1,"{2706, 2606, 30, 8}"
50,1,51,0,0.0889617,"\int \frac{\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d}","-\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d}",1,"-Cot[c + d*x]^4/(4*a*d) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(3*a*d)","A",5,4,21,0.1905,1,"{2706, 2607, 30, 2606}"
51,1,68,0,0.0933341,"\int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d}","-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d}",1,"-Cot[c + d*x]^6/(6*a*d) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
52,1,84,0,0.0997991,"\int \frac{\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^9/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{3 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}","-\frac{\cot ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{3 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}",1,"-Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(a*d) - (3*Csc[c + d*x]^5)/(5*a*d) + Csc[c + d*x]^7/(7*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
53,1,84,0,0.0973616,"\int \frac{\tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^6/(a + a*Sin[c + d*x]),x]","\frac{\tan ^7(c+d x)}{7 a d}-\frac{\sec ^7(c+d x)}{7 a d}+\frac{3 \sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{a d}+\frac{\sec (c+d x)}{a d}","\frac{\tan ^7(c+d x)}{7 a d}-\frac{\sec ^7(c+d x)}{7 a d}+\frac{3 \sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{a d}+\frac{\sec (c+d x)}{a d}",1,"Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^7/(7*a*d) + Tan[c + d*x]^7/(7*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
54,1,69,0,0.0918369,"\int \frac{\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x]),x]","\frac{\tan ^5(c+d x)}{5 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{2 \sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}","\frac{\tan ^5(c+d x)}{5 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{2 \sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}",1,"-(Sec[c + d*x]/(a*d)) + (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^5/(5*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
55,1,50,0,0.0876481,"\int \frac{\tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}","\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}",1,"Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]^3/(3*a*d)","A",5,4,21,0.1905,1,"{2706, 2607, 30, 2606}"
56,1,23,0,0.0116354,"\int \frac{1}{a+a \sin (c+d x)} \, dx","Int[(a + a*Sin[c + d*x])^(-1),x]","-\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}","-\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}",1,"-(Cos[c + d*x]/(d*(a + a*Sin[c + d*x])))","A",1,1,12,0.08333,1,"{2648}"
57,1,29,0,0.0512723,"\int \frac{\cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}",1,"ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d)","A",4,4,21,0.1905,1,"{2706, 3767, 8, 3770}"
58,1,58,0,0.088263,"\int \frac{\cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^3(c+d x)}{3 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{\cot ^3(c+d x)}{3 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]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
59,1,82,0,0.10552,"\int \frac{\cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}","-\frac{\cot ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}",1,"(3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]^5/(5*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
60,1,106,0,0.127459,"\int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}","-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(16*a*d) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)","A",7,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
61,1,189,0,0.1486046,"\int \frac{\tan ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]","\frac{a^3}{80 d (a \sin (c+d x)+a)^5}-\frac{5 a^2}{64 d (a \sin (c+d x)+a)^4}+\frac{21}{256 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{35}{256 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac{a}{192 d (a-a \sin (c+d x))^3}+\frac{19 a}{96 d (a \sin (c+d x)+a)^3}-\frac{1}{32 d (a-a \sin (c+d x))^2}-\frac{1}{4 d (a \sin (c+d x)+a)^2}","\frac{a^3}{80 d (a \sin (c+d x)+a)^5}-\frac{5 a^2}{64 d (a \sin (c+d x)+a)^4}+\frac{21}{256 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{35}{256 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac{a}{192 d (a-a \sin (c+d x))^3}+\frac{19 a}{96 d (a \sin (c+d x)+a)^3}-\frac{1}{32 d (a-a \sin (c+d x))^2}-\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"(-7*ArcTanh[Sin[c + d*x]])/(128*a^2*d) + a/(192*d*(a - a*Sin[c + d*x])^3) - 1/(32*d*(a - a*Sin[c + d*x])^2) + a^3/(80*d*(a + a*Sin[c + d*x])^5) - (5*a^2)/(64*d*(a + a*Sin[c + d*x])^4) + (19*a)/(96*d*(a + a*Sin[c + d*x])^3) - 1/(4*d*(a + a*Sin[c + d*x])^2) + 21/(256*d*(a^2 - a^2*Sin[c + d*x])) + 35/(256*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
62,1,146,0,0.1079584,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{a^2}{32 d (a \sin (c+d x)+a)^4}-\frac{5}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{7 a}{48 d (a \sin (c+d x)+a)^3}+\frac{1}{64 d (a-a \sin (c+d x))^2}+\frac{1}{4 d (a \sin (c+d x)+a)^2}","\frac{a^2}{32 d (a \sin (c+d x)+a)^4}-\frac{5}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{7 a}{48 d (a \sin (c+d x)+a)^3}+\frac{1}{64 d (a-a \sin (c+d x))^2}+\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"(5*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + 1/(64*d*(a - a*Sin[c + d*x])^2) + a^2/(32*d*(a + a*Sin[c + d*x])^4) - (7*a)/(48*d*(a + a*Sin[c + d*x])^3) + 1/(4*d*(a + a*Sin[c + d*x])^2) - 5/(64*d*(a^2 - a^2*Sin[c + d*x])) - 5/(32*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
63,1,104,0,0.085784,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","\frac{1}{16 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{3}{16 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^2 d}+\frac{a}{12 d (a \sin (c+d x)+a)^3}-\frac{1}{4 d (a \sin (c+d x)+a)^2}","\frac{1}{16 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{3}{16 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^2 d}+\frac{a}{12 d (a \sin (c+d x)+a)^3}-\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"-ArcTanh[Sin[c + d*x]]/(8*a^2*d) + a/(12*d*(a + a*Sin[c + d*x])^3) - 1/(4*d*(a + a*Sin[c + d*x])^2) + 1/(16*d*(a^2 - a^2*Sin[c + d*x])) + 3/(16*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
64,1,60,0,0.0483769,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}+\frac{1}{4 d (a \sin (c+d x)+a)^2}","-\frac{1}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}+\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(4*a^2*d) + 1/(4*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2707, 77, 206}"
65,1,52,0,0.0504909,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d}","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d}",1,"Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 44}"
66,1,65,0,0.0593806,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{2 \log (\sin (c+d x))}{a^2 d}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d}","-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{2 \log (\sin (c+d x))}{a^2 d}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d}",1,"(2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + (2*Log[Sin[c + d*x]])/(a^2*d) - (2*Log[1 + Sin[c + d*x]])/(a^2*d)","A",3,2,21,0.09524,1,"{2707, 77}"
67,1,55,0,0.0511339,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"-Csc[c + d*x]^2/(2*a^2*d) + (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a^2*d)","A",3,2,21,0.09524,1,"{2707, 43}"
68,1,73,0,0.0571898,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^6(c+d x)}{6 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{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^6(c+d x)}{6 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{\csc ^2(c+d x)}{2 a^2 d}",1,"Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a^2*d)","A",3,2,21,0.09524,1,"{2707, 75}"
69,1,127,0,0.0738176,"\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^8(c+d x)}{8 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}+\frac{\csc ^6(c+d x)}{6 a^2 d}-\frac{4 \csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^8(c+d x)}{8 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}+\frac{\csc ^6(c+d x)}{6 a^2 d}-\frac{4 \csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"-Csc[c + d*x]^2/(2*a^2*d) + (2*Csc[c + d*x]^3)/(3*a^2*d) + Csc[c + d*x]^4/(4*a^2*d) - (4*Csc[c + d*x]^5)/(5*a^2*d) + Csc[c + d*x]^6/(6*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d) - Csc[c + d*x]^8/(8*a^2*d)","A",3,2,21,0.09524,1,"{2707, 88}"
70,1,145,0,0.0811955,"\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^{10}(c+d x)}{10 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}+\frac{\csc ^8(c+d x)}{4 a^2 d}-\frac{6 \csc ^7(c+d x)}{7 a^2 d}+\frac{6 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(2*a^2*d) + (6*Csc[c + d*x]^5)/(5*a^2*d) - (6*Csc[c + d*x]^7)/(7*a^2*d) + Csc[c + d*x]^8/(4*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d) - Csc[c + d*x]^10/(10*a^2*d)","A",3,2,21,0.09524,1,"{2707, 88}"
71,1,199,0,0.1023942,"\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^{12}(c+d x)}{12 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}+\frac{3 \csc ^{10}(c+d x)}{10 a^2 d}-\frac{8 \csc ^9(c+d x)}{9 a^2 d}-\frac{\csc ^8(c+d x)}{4 a^2 d}+\frac{12 \csc ^7(c+d x)}{7 a^2 d}-\frac{\csc ^6(c+d x)}{3 a^2 d}-\frac{8 \csc ^5(c+d x)}{5 a^2 d}+\frac{3 \csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^{12}(c+d x)}{12 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}+\frac{3 \csc ^{10}(c+d x)}{10 a^2 d}-\frac{8 \csc ^9(c+d x)}{9 a^2 d}-\frac{\csc ^8(c+d x)}{4 a^2 d}+\frac{12 \csc ^7(c+d x)}{7 a^2 d}-\frac{\csc ^6(c+d x)}{3 a^2 d}-\frac{8 \csc ^5(c+d x)}{5 a^2 d}+\frac{3 \csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"-Csc[c + d*x]^2/(2*a^2*d) + (2*Csc[c + d*x]^3)/(3*a^2*d) + (3*Csc[c + d*x]^4)/(4*a^2*d) - (8*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(3*a^2*d) + (12*Csc[c + d*x]^7)/(7*a^2*d) - Csc[c + d*x]^8/(4*a^2*d) - (8*Csc[c + d*x]^9)/(9*a^2*d) + (3*Csc[c + d*x]^10)/(10*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d) - Csc[c + d*x]^12/(12*a^2*d)","A",3,2,21,0.09524,1,"{2707, 88}"
72,1,171,0,0.1233514,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{a^2}{40 d (a \sin (c+d x)+a)^5}-\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{5}{128 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{7 a}{64 d (a \sin (c+d x)+a)^4}+\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2}","\frac{a^2}{40 d (a \sin (c+d x)+a)^5}-\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{5}{128 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{7 a}{64 d (a \sin (c+d x)+a)^4}+\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(128*a^3*d) + 1/(128*a*d*(a - a*Sin[c + d*x])^2) + a^2/(40*d*(a + a*Sin[c + d*x])^5) - (7*a)/(64*d*(a + a*Sin[c + d*x])^4) + 1/(6*d*(a + a*Sin[c + d*x])^3) - 5/(64*a*d*(a + a*Sin[c + d*x])^2) - 1/(32*d*(a^3 - a^3*Sin[c + d*x])) - 5/(128*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
73,1,126,0,0.090512,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{1}{16 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\tanh ^{-1}(\sin (c+d x))}{32 a^3 d}+\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{3}{32 a d (a \sin (c+d x)+a)^2}","\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{1}{16 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\tanh ^{-1}(\sin (c+d x))}{32 a^3 d}+\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{3}{32 a d (a \sin (c+d x)+a)^2}",1,"-ArcTanh[Sin[c + d*x]]/(32*a^3*d) + a/(16*d*(a + a*Sin[c + d*x])^4) - 1/(6*d*(a + a*Sin[c + d*x])^3) + 3/(32*a*d*(a + a*Sin[c + d*x])^2) + 1/(32*d*(a^3 - a^3*Sin[c + d*x])) + 1/(16*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
74,1,82,0,0.0571133,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]/(a + a*Sin[c + d*x])^3,x]","-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{1}{8 a d (a \sin (c+d x)+a)^2}+\frac{1}{6 d (a \sin (c+d x)+a)^3}","-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{1}{8 a d (a \sin (c+d x)+a)^2}+\frac{1}{6 d (a \sin (c+d x)+a)^3}",1,"ArcTanh[Sin[c + d*x]]/(8*a^3*d) + 1/(6*d*(a + a*Sin[c + d*x])^3) - 1/(8*a*d*(a + a*Sin[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2707, 77, 206}"
75,1,74,0,0.0582604,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{1}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{1}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"Log[Sin[c + d*x]]/(a^3*d) - Log[1 + Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 1/(d*(a^3 + a^3*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 44}"
76,1,86,0,0.0697399,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{5 \log (\sin (c+d x))}{a^3 d}-\frac{5 \log (\sin (c+d x)+1)}{a^3 d}","\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{5 \log (\sin (c+d x))}{a^3 d}-\frac{5 \log (\sin (c+d x)+1)}{a^3 d}",1,"(3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (5*Log[Sin[c + d*x]])/(a^3*d) - (5*Log[1 + Sin[c + d*x]])/(a^3*d) + 2/(d*(a^3 + a^3*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 77}"
77,1,96,0,0.0712619,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(4*Csc[c + d*x])/(a^3*d) - (2*Csc[c + d*x]^2)/(a^3*d) + Csc[c + d*x]^3/(a^3*d) - Csc[c + d*x]^4/(4*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",3,2,21,0.09524,1,"{2707, 88}"
78,1,73,0,0.0574368,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d}","-\frac{\csc ^6(c+d x)}{6 a^3 d}+\frac{3 \csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d}",1,"Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + (3*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(6*a^3*d)","A",3,2,21,0.09524,1,"{2707, 43}"
79,1,109,0,0.0689768,"\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^8(c+d x)}{8 a^3 d}+\frac{3 \csc ^7(c+d x)}{7 a^3 d}-\frac{\csc ^6(c+d x)}{3 a^3 d}-\frac{2 \csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d}","-\frac{\csc ^8(c+d x)}{8 a^3 d}+\frac{3 \csc ^7(c+d x)}{7 a^3 d}-\frac{\csc ^6(c+d x)}{3 a^3 d}-\frac{2 \csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d}",1,"-Csc[c + d*x]^3/(3*a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (2*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(3*a^3*d) + (3*Csc[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^8/(8*a^3*d)","A",3,2,21,0.09524,1,"{2707, 75}"
80,1,145,0,0.0805763,"\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^{10}(c+d x)}{10 a^3 d}+\frac{\csc ^9(c+d x)}{3 a^3 d}-\frac{\csc ^8(c+d x)}{8 a^3 d}-\frac{5 \csc ^7(c+d x)}{7 a^3 d}+\frac{5 \csc ^6(c+d x)}{6 a^3 d}+\frac{\csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d}","-\frac{\csc ^{10}(c+d x)}{10 a^3 d}+\frac{\csc ^9(c+d x)}{3 a^3 d}-\frac{\csc ^8(c+d x)}{8 a^3 d}-\frac{5 \csc ^7(c+d x)}{7 a^3 d}+\frac{5 \csc ^6(c+d x)}{6 a^3 d}+\frac{\csc ^5(c+d x)}{5 a^3 d}-\frac{3 \csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{3 a^3 d}",1,"Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + Csc[c + d*x]^5/(5*a^3*d) + (5*Csc[c + d*x]^6)/(6*a^3*d) - (5*Csc[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^8/(8*a^3*d) + Csc[c + d*x]^9/(3*a^3*d) - Csc[c + d*x]^10/(10*a^3*d)","A",3,2,21,0.09524,1,"{2707, 88}"
81,1,145,0,0.0783676,"\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^{12}(c+d x)}{12 a^3 d}+\frac{3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{8 \csc ^9(c+d x)}{9 a^3 d}+\frac{3 \csc ^8(c+d x)}{4 a^3 d}+\frac{6 \csc ^7(c+d x)}{7 a^3 d}-\frac{4 \csc ^6(c+d x)}{3 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d}","-\frac{\csc ^{12}(c+d x)}{12 a^3 d}+\frac{3 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{8 \csc ^9(c+d x)}{9 a^3 d}+\frac{3 \csc ^8(c+d x)}{4 a^3 d}+\frac{6 \csc ^7(c+d x)}{7 a^3 d}-\frac{4 \csc ^6(c+d x)}{3 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d}",1,"-Csc[c + d*x]^3/(3*a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (4*Csc[c + d*x]^6)/(3*a^3*d) + (6*Csc[c + d*x]^7)/(7*a^3*d) + (3*Csc[c + d*x]^8)/(4*a^3*d) - (8*Csc[c + d*x]^9)/(9*a^3*d) + (3*Csc[c + d*x]^11)/(11*a^3*d) - Csc[c + d*x]^12/(12*a^3*d)","A",3,2,21,0.09524,1,"{2707, 88}"
82,1,195,0,0.134569,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]","\frac{a^2}{48 d (a \sin (c+d x)+a)^6}-\frac{3}{256 d \left(a^4-a^4 \sin (c+d x)\right)}-\frac{1}{256 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{256 d \left(a^2-a^2 \sin (c+d x)\right)^2}-\frac{5}{256 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}-\frac{7 a}{80 d (a \sin (c+d x)+a)^5}+\frac{1}{8 d (a \sin (c+d x)+a)^4}-\frac{5}{96 a d (a \sin (c+d x)+a)^3}","\frac{a^2}{48 d (a \sin (c+d x)+a)^6}-\frac{3}{256 d \left(a^4-a^4 \sin (c+d x)\right)}-\frac{1}{256 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{256 d \left(a^2-a^2 \sin (c+d x)\right)^2}-\frac{5}{256 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}-\frac{7 a}{80 d (a \sin (c+d x)+a)^5}+\frac{1}{8 d (a \sin (c+d x)+a)^4}-\frac{5}{96 a d (a \sin (c+d x)+a)^3}",1,"-ArcTanh[Sin[c + d*x]]/(128*a^4*d) + a^2/(48*d*(a + a*Sin[c + d*x])^6) - (7*a)/(80*d*(a + a*Sin[c + d*x])^5) + 1/(8*d*(a + a*Sin[c + d*x])^4) - 5/(96*a*d*(a + a*Sin[c + d*x])^3) + 1/(256*d*(a^2 - a^2*Sin[c + d*x])^2) - 5/(256*d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(256*d*(a^4 - a^4*Sin[c + d*x])) - 1/(256*d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2707, 88, 206}"
83,1,132,0,0.0887882,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^4,x]","\frac{1}{64 d \left(a^4-a^4 \sin (c+d x)\right)}+\frac{1}{64 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{32 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{a}{20 d (a \sin (c+d x)+a)^5}-\frac{1}{8 d (a \sin (c+d x)+a)^4}+\frac{1}{16 a d (a \sin (c+d x)+a)^3}","\frac{1}{64 d \left(a^4-a^4 \sin (c+d x)\right)}+\frac{1}{64 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{32 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{a}{20 d (a \sin (c+d x)+a)^5}-\frac{1}{8 d (a \sin (c+d x)+a)^4}+\frac{1}{16 a d (a \sin (c+d x)+a)^3}",1,"a/(20*d*(a + a*Sin[c + d*x])^5) - 1/(8*d*(a + a*Sin[c + d*x])^4) + 1/(16*a*d*(a + a*Sin[c + d*x])^3) + 1/(32*d*(a^2 + a^2*Sin[c + d*x])^2) + 1/(64*d*(a^4 - a^4*Sin[c + d*x])) + 1/(64*d*(a^4 + a^4*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
84,1,105,0,0.0653541,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Tan[c + d*x]/(a + a*Sin[c + d*x])^4,x]","-\frac{1}{16 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{1}{16 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{16 a^4 d}-\frac{1}{12 a d (a \sin (c+d x)+a)^3}+\frac{1}{8 d (a \sin (c+d x)+a)^4}","-\frac{1}{16 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{1}{16 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{16 a^4 d}-\frac{1}{12 a d (a \sin (c+d x)+a)^3}+\frac{1}{8 d (a \sin (c+d x)+a)^4}",1,"ArcTanh[Sin[c + d*x]]/(16*a^4*d) + 1/(8*d*(a + a*Sin[c + d*x])^4) - 1/(12*a*d*(a + a*Sin[c + d*x])^3) - 1/(16*d*(a^2 + a^2*Sin[c + d*x])^2) - 1/(16*d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2707, 77, 206}"
85,1,106,0,0.082628,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^4,x]","\frac{5}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{9 \log (\sin (c+d x))}{a^4 d}-\frac{9 \log (\sin (c+d x)+1)}{a^4 d}","\frac{5}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{9 \log (\sin (c+d x))}{a^4 d}-\frac{9 \log (\sin (c+d x)+1)}{a^4 d}",1,"(4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (9*Log[Sin[c + d*x]])/(a^4*d) - (9*Log[1 + Sin[c + d*x]])/(a^4*d) + 1/(d*(a^2 + a^2*Sin[c + d*x])^2) + 5/(d*(a^4 + a^4*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 77}"
86,1,135,0,0.0856235,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]","-\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d}","-\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d}",1,"(8*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (8*Csc[c + d*x]^3)/(3*a^4*d) - (7*Csc[c + d*x]^4)/(4*a^4*d) + (4*Csc[c + d*x]^5)/(5*a^4*d) - Csc[c + d*x]^6/(6*a^4*d) + (8*Log[Sin[c + d*x]])/(a^4*d) - (8*Log[1 + Sin[c + d*x]])/(a^4*d)","A",3,2,21,0.09524,1,"{2707, 88}"
87,1,127,0,0.309682,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^4,x]","\frac{8 \tan ^9(c+d x)}{9 a^4 d}+\frac{16 \tan ^7(c+d x)}{7 a^4 d}+\frac{9 \tan ^5(c+d x)}{5 a^4 d}+\frac{\tan ^3(c+d x)}{3 a^4 d}-\frac{8 \sec ^9(c+d x)}{9 a^4 d}+\frac{12 \sec ^7(c+d x)}{7 a^4 d}-\frac{4 \sec ^5(c+d x)}{5 a^4 d}","\frac{8 \tan ^9(c+d x)}{9 a^4 d}+\frac{16 \tan ^7(c+d x)}{7 a^4 d}+\frac{9 \tan ^5(c+d x)}{5 a^4 d}+\frac{\tan ^3(c+d x)}{3 a^4 d}-\frac{8 \sec ^9(c+d x)}{9 a^4 d}+\frac{12 \sec ^7(c+d x)}{7 a^4 d}-\frac{4 \sec ^5(c+d x)}{5 a^4 d}",1,"(-4*Sec[c + d*x]^5)/(5*a^4*d) + (12*Sec[c + d*x]^7)/(7*a^4*d) - (8*Sec[c + d*x]^9)/(9*a^4*d) + Tan[c + d*x]^3/(3*a^4*d) + (9*Tan[c + d*x]^5)/(5*a^4*d) + (16*Tan[c + d*x]^7)/(7*a^4*d) + (8*Tan[c + d*x]^9)/(9*a^4*d)","A",17,5,21,0.2381,1,"{2711, 2607, 270, 2606, 14}"
88,1,108,0,0.3179347,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^4,x]","-\frac{\cot (c+d x)}{a^4 d}+\frac{4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{104 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)}+\frac{31 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)^2}-\frac{2 \cot (c+d x)}{5 a^4 d (\csc (c+d x)+1)^3}","-\frac{\cot (c+d x)}{a^4 d}+\frac{4 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{104 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)}+\frac{31 \cot (c+d x)}{15 a^4 d (\csc (c+d x)+1)^2}-\frac{2 \cot (c+d x)}{5 a^4 d (\csc (c+d x)+1)^3}",1,"(4*ArcTanh[Cos[c + d*x]])/(a^4*d) - Cot[c + d*x]/(a^4*d) - (2*Cot[c + d*x])/(5*a^4*d*(1 + Csc[c + d*x])^3) + (31*Cot[c + d*x])/(15*a^4*d*(1 + Csc[c + d*x])^2) - (104*Cot[c + d*x])/(15*a^4*d*(1 + Csc[c + d*x]))","A",14,8,21,0.3810,1,"{2709, 3770, 3767, 8, 3777, 3922, 3919, 3794}"
89,1,120,0,0.248752,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]","-\frac{\cot ^3(c+d x)}{3 a^4 d}-\frac{9 \cot (c+d x)}{a^4 d}+\frac{14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 \cot (c+d x) \csc (c+d x)}{a^4 d}-\frac{44 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)}+\frac{4 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)^2}","-\frac{\cot ^3(c+d x)}{3 a^4 d}-\frac{9 \cot (c+d x)}{a^4 d}+\frac{14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 \cot (c+d x) \csc (c+d x)}{a^4 d}-\frac{44 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)}+\frac{4 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)^2}",1,"(14*ArcTanh[Cos[c + d*x]])/(a^4*d) - (9*Cot[c + d*x])/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + (2*Cot[c + d*x]*Csc[c + d*x])/(a^4*d) + (4*Cot[c + d*x])/(3*a^4*d*(1 + Csc[c + d*x])^2) - (44*Cot[c + d*x])/(3*a^4*d*(1 + Csc[c + d*x]))","A",14,8,21,0.3810,1,"{2709, 3770, 3767, 8, 3768, 3777, 3919, 3794}"
90,1,133,0,0.2493068,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]","-\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}+\frac{27 \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (\csc (c+d x)+1)}","-\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}+\frac{27 \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (\csc (c+d x)+1)}",1,"(27*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - (16*Cot[c + d*x])/(a^4*d) - (3*Cot[c + d*x]^3)/(a^4*d) - Cot[c + d*x]^5/(5*a^4*d) + (11*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(a^4*d) - (8*Cot[c + d*x])/(a^4*d*(1 + Csc[c + d*x]))","A",16,6,21,0.2857,1,"{2709, 3770, 3767, 8, 3768, 3777}"
91,1,195,0,0.9230522,"\int \sqrt{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*Tan[e + f*x]^4,x]","\frac{11 a^2 \cos (e+f x)}{8 f (a \sin (e+f x)+a)^{3/2}}-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}+\frac{4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 a f}-\frac{7 \sec ^3(e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{11 a \sec (e+f x)}{6 f \sqrt{a \sin (e+f x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{8 \sqrt{2} f}","\frac{5 \tan ^3(e+f x) \sqrt{a (\sin (e+f x)+1)}}{12 f}+\frac{29 \tan (e+f x) \sqrt{a \sin (e+f x)+a}}{12 f}-\frac{\sec ^3(e+f x) \sqrt{a (\sin (e+f x)+1)}}{12 f}-\frac{27 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)}}{8 f}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{8 \sqrt{2} f}",1,"(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(8*Sqrt[2]*f) + (11*a^2*Cos[e + f*x])/(8*f*(a + a*Sin[e + f*x])^(3/2)) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (11*a*Sec[e + f*x])/(6*f*Sqrt[a + a*Sin[e + f*x]]) - (7*Sec[e + f*x]^3*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*a*f)","A",15,10,23,0.4348,1,"{2714, 2646, 4401, 2675, 2687, 2650, 2649, 206, 2878, 2855}"
92,1,101,0,0.1847536,"\int \sqrt{a+a \sin (e+f x)} \tan ^2(e+f x) \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*Tan[e + f*x]^2,x]","-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{a f}+\frac{5 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{2} f}","-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{a f}+\frac{5 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{2} f}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*f)) + (5*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(a*f)","A",4,4,23,0.1739,1,"{2713, 2855, 2649, 206}"
93,1,89,0,0.1924125,"\int \cot ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Int[Cot[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]],x]","\frac{3 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{\cot (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}","\frac{3 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{\cot (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (3*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f","A",4,4,23,0.1739,1,"{2716, 2981, 2773, 206}"
94,1,163,0,0.37672,"\int \cot ^4(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Int[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]","-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}}","-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}}",1,"(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) + (11*a*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f)","A",7,7,23,0.3043,1,"{2718, 2646, 3044, 2980, 2772, 2773, 206}"
95,1,195,0,0.9750158,"\int (a+a \sin (e+f x))^{3/2} \tan ^4(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^4,x]","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}+\frac{4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{5/2}}{a f}-\frac{23 \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}+\frac{a \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{2 f}","\frac{2 a^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{3/2}}-\frac{4 a^2 \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} f}+\frac{\sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac{7 a \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{2 f}",1,"-(a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*f) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (a*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*f) - (23*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f) + (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(5/2))/(a*f)","A",14,9,23,0.3913,1,"{2714, 2647, 2646, 4401, 2675, 2649, 206, 2878, 2855}"
96,1,88,0,0.1957312,"\int (a+a \sin (e+f x))^{3/2} \tan ^2(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^2,x]","\frac{11 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f}+\frac{7 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}","\frac{11 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f}+\frac{7 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}",1,"(11*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (7*Sec[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(3*f) - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f)","A",3,3,23,0.1304,1,"{2713, 2855, 2646}"
97,1,121,0,0.3205684,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Int[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2),x]","\frac{11 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}+\frac{5 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}","\frac{11 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}+\frac{5 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}",1,"(-3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f + (11*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (5*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (Cot[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/f","A",5,5,23,0.2174,1,"{2716, 2976, 2981, 2773, 206}"
98,1,197,0,0.4963622,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a \sin (e+f x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f}","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{29 a^2 \cot (e+f x)}{24 f \sqrt{a \sin (e+f x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f}",1,"(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (29*a^2*Cot[e + f*x])/(24*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(3*f)","A",8,8,23,0.3478,1,"{2718, 2647, 2646, 3044, 2975, 2980, 2773, 206}"
99,1,208,0,0.9790985,"\int (a+a \sin (e+f x))^{5/2} \tan ^4(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*Tan[e + f*x]^4,x]","-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{64 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{46 a^2 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{7/2}}{a f}+\frac{26 \sec ^3(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac{2 a \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}","-\frac{2 a^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^{5/2}}+\frac{8 a^4 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{3/2}}-\frac{12 a^3 \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{f}+\frac{2 a \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}",1,"(-64*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (46*a^2*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (2*a*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f) + (26*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(5/2))/(3*f) - (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(7/2))/(a*f)","A",10,7,23,0.3043,1,"{2714, 2647, 2646, 4401, 2673, 2878, 2855}"
100,1,118,0,0.2142872,"\int (a+a \sin (e+f x))^{5/2} \tan ^2(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*Tan[e + f*x]^2,x]","\frac{31 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{124 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{7/2}}{5 a f}+\frac{9 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{5 f}","\frac{31 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{124 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 \sec (e+f x) (a \sin (e+f x)+a)^{7/2}}{5 a f}+\frac{9 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{5 f}",1,"(124*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) + (31*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (9*Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(5*f) - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(5*a*f)","A",4,4,23,0.1739,1,"{2713, 2855, 2647, 2646}"
101,1,151,0,0.4286961,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^{5/2} \, dx","Int[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2),x]","\frac{49 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}+\frac{31 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}+\frac{7 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) (a \sin (e+f x)+a)^{5/2}}{f}","\frac{49 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}+\frac{31 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}+\frac{7 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) (a \sin (e+f x)+a)^{5/2}}{f}",1,"(-5*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f + (49*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) + (31*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (7*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (Cot[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/f","A",6,5,23,0.2174,1,"{2716, 2976, 2981, 2773, 206}"
102,1,227,0,0.6271512,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx","Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]","-\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a \sin (e+f x)+a}}{24 f}+\frac{55 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{12 f}","-\frac{9 a^3 \cos (e+f x)}{40 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}+\frac{17 a^2 \cot (e+f x) \sqrt{a \sin (e+f x)+a}}{24 f}+\frac{55 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac{\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac{5 a \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{12 f}",1,"(55*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (9*a^3*Cos[e + f*x])/(40*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (17*a^2*Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(24*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (5*a*Cot[e + f*x]*Csc[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(12*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(3*f)","A",10,8,23,0.3478,1,"{2718, 2647, 2646, 3044, 2975, 2981, 2773, 206}"
103,1,241,0,0.9333506,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Tan[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]],x]","\frac{61 a \cos (e+f x)}{64 f (a \sin (e+f x)+a)^{3/2}}+\frac{7 \sec ^3(e+f x) \sqrt{a \sin (e+f x)+a}}{12 a f}-\frac{5 \sec ^3(e+f x)}{6 f \sqrt{a \sin (e+f x)+a}}-\frac{61 \sec (e+f x)}{48 f \sqrt{a \sin (e+f x)+a}}+\frac{7 a \sec (e+f x)}{24 f (a \sin (e+f x)+a)^{3/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}+\frac{61 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{64 \sqrt{2} \sqrt{a} f}","\frac{\tan ^3(e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{a \sin (e+f x) \tan (e+f x)}{24 f (a \sin (e+f x)+a)^{3/2}}-\frac{(127 \sin (e+f x)+53) \sec (e+f x)}{192 f \sqrt{a \sin (e+f x)+a}}-\frac{67 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{64 \sqrt{2} \sqrt{a} f}",1,"(61*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(64*Sqrt[2]*Sqrt[a]*f) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f) + (61*a*Cos[e + f*x])/(64*f*(a + a*Sin[e + f*x])^(3/2)) + (7*a*Sec[e + f*x])/(24*f*(a + a*Sin[e + f*x])^(3/2)) - (61*Sec[e + f*x])/(48*f*Sqrt[a + a*Sin[e + f*x]]) - (5*Sec[e + f*x]^3)/(6*f*Sqrt[a + a*Sin[e + f*x]]) + (7*Sec[e + f*x]^3*Sqrt[a + a*Sin[e + f*x]])/(12*a*f)","A",17,9,23,0.3913,1,"{2714, 2649, 206, 4401, 2687, 2681, 2650, 2877, 2855}"
104,1,107,0,0.1952455,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Tan[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]],x]","\frac{3 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f}-\frac{\sec (e+f x)}{2 f \sqrt{a \sin (e+f x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{2} \sqrt{a} f}","\frac{3 \sec (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f}-\frac{\sec (e+f x)}{2 f \sqrt{a \sin (e+f x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{2} \sqrt{a} f}",1,"(5*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[2]*Sqrt[a]*f) - Sec[e + f*x]/(2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f)","A",4,4,23,0.1739,1,"{2712, 2855, 2649, 206}"
105,1,62,0,0.1100945,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Cot[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{\cot (e+f x)}{f \sqrt{a \sin (e+f x)+a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{\cot (e+f x)}{f \sqrt{a \sin (e+f x)+a}}",1,"ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]]/(Sqrt[a]*f) - Cot[e + f*x]/(f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,23,0.1739,1,"{2716, 21, 2773, 206}"
106,1,135,0,0.621613,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Cot[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]],x]","\frac{9 \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{\cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}}","\frac{9 \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\frac{\cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}}",1,"(-7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*Sqrt[a]*f) + (9*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) + (Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a + a*Sin[e + f*x]])","A",11,7,23,0.3043,1,"{2718, 2649, 206, 3044, 2984, 2985, 2773}"
107,1,195,0,1.1992705,"\int \frac{\tan ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2),x]","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{256 \sqrt{2} a^{3/2} f}+\frac{7 \cos (e+f x)}{256 f (a \sin (e+f x)+a)^{3/2}}+\frac{\sec ^3(e+f x)}{4 a f \sqrt{a \sin (e+f x)+a}}-\frac{\sec ^3(e+f x)}{6 f (a \sin (e+f x)+a)^{3/2}}-\frac{45 \sec (e+f x)}{64 a f \sqrt{a \sin (e+f x)+a}}+\frac{9 \sec (e+f x)}{32 f (a \sin (e+f x)+a)^{3/2}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{256 \sqrt{2} a^{3/2} f}+\frac{\tan ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{3/2}}+\frac{a \sin (e+f x) \tan (e+f x)}{12 f (a \sin (e+f x)+a)^{5/2}}+\frac{7 \cos (e+f x)}{256 f (a \sin (e+f x)+a)^{3/2}}-\frac{(87 \sin (e+f x)+65) \sec (e+f x)}{192 f (a \sin (e+f x)+a)^{3/2}}",1,"(7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(256*Sqrt[2]*a^(3/2)*f) + (7*Cos[e + f*x])/(256*f*(a + a*Sin[e + f*x])^(3/2)) + (9*Sec[e + f*x])/(32*f*(a + a*Sin[e + f*x])^(3/2)) - Sec[e + f*x]^3/(6*f*(a + a*Sin[e + f*x])^(3/2)) - (45*Sec[e + f*x])/(64*a*f*Sqrt[a + a*Sin[e + f*x]]) + Sec[e + f*x]^3/(4*a*f*Sqrt[a + a*Sin[e + f*x]])","A",20,9,23,0.3913,1,"{2714, 2650, 2649, 206, 4401, 2681, 2687, 2877, 2855}"
108,1,134,0,0.2232659,"\int \frac{\tan ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{32 \sqrt{2} a^{3/2} f}+\frac{\cos (e+f x)}{32 f (a \sin (e+f x)+a)^{3/2}}+\frac{5 \sec (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}-\frac{\sec (e+f x)}{4 f (a \sin (e+f x)+a)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{32 \sqrt{2} a^{3/2} f}+\frac{\cos (e+f x)}{32 f (a \sin (e+f x)+a)^{3/2}}+\frac{5 \sec (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}-\frac{\sec (e+f x)}{4 f (a \sin (e+f x)+a)^{3/2}}",1,"ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(32*Sqrt[2]*a^(3/2)*f) + Cos[e + f*x]/(32*f*(a + a*Sin[e + f*x])^(3/2)) - Sec[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(3/2)) + (5*Sec[e + f*x])/(8*a*f*Sqrt[a + a*Sin[e + f*x]])","A",5,5,23,0.2174,1,"{2712, 2855, 2650, 2649, 206}"
109,1,113,0,0.2288058,"\int \frac{\cot ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f \sqrt{a \sin (e+f x)+a}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f \sqrt{a \sin (e+f x)+a}}",1,"(3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(a^(3/2)*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*f) - Cot[e + f*x]/(a*f*Sqrt[a + a*Sin[e + f*x]])","A",6,5,23,0.2174,1,"{2715, 2985, 2649, 206, 2773}"
110,1,144,0,0.5531068,"\int \frac{\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 a^2 f}-\frac{\cot (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a \sin (e+f x)+a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 a^2 f}-\frac{\cot (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a}}+\frac{11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt{a \sin (e+f x)+a}}",1,"-ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]]/(8*a^(3/2)*f) - Cot[e + f*x]/(8*a*f*Sqrt[a + a*Sin[e + f*x]]) + (11*Cot[e + f*x]*Csc[e + f*x])/(12*a*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*a^2*f)","A",10,6,23,0.2609,1,"{2717, 2772, 2773, 206, 3044, 2980}"
111,1,260,0,1.4337388,"\int \frac{\tan ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{31 \sec ^3(e+f x)}{192 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{1085 \sec (e+f x)}{3072 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{317 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{4096 \sqrt{2} a^{5/2} f}+\frac{317 \cos (e+f x)}{4096 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}+\frac{53 \sec ^3(e+f x)}{96 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\sec ^3(e+f x)}{8 f (a \sin (e+f x)+a)^{5/2}}+\frac{217 \sec (e+f x)}{1536 a f (a \sin (e+f x)+a)^{3/2}}","\frac{317 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{4096 \sqrt{2} a^{5/2} f}+\frac{\tan ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{5/2}}+\frac{5 a \sin (e+f x) \tan (e+f x)}{48 f (a \sin (e+f x)+a)^{7/2}}+\frac{317 \cos (e+f x)}{4096 a f (a \sin (e+f x)+a)^{3/2}}+\frac{317 \cos (e+f x)}{3072 f (a \sin (e+f x)+a)^{5/2}}-\frac{(129 \sin (e+f x)+115) \sec (e+f x)}{384 f (a \sin (e+f x)+a)^{5/2}}",1,"(317*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(4096*Sqrt[2]*a^(5/2)*f) - Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)) - Sec[e + f*x]^3/(8*f*(a + a*Sin[e + f*x])^(5/2)) + (317*Cos[e + f*x])/(4096*a*f*(a + a*Sin[e + f*x])^(3/2)) + (217*Sec[e + f*x])/(1536*a*f*(a + a*Sin[e + f*x])^(3/2)) + (53*Sec[e + f*x]^3)/(96*a*f*(a + a*Sin[e + f*x])^(3/2)) - (1085*Sec[e + f*x])/(3072*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (31*Sec[e + f*x]^3)/(192*a^2*f*Sqrt[a + a*Sin[e + f*x]])","A",23,9,23,0.3913,1,"{2714, 2650, 2649, 206, 4401, 2681, 2687, 2877, 2859}"
112,1,167,0,0.3028203,"\int \frac{\tan ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2),x]","\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{128 \sqrt{2} a^{5/2} f}-\frac{11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}}","\frac{11 \sec (e+f x)}{96 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{128 \sqrt{2} a^{5/2} f}-\frac{11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac{17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}}",1,"(-11*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(128*Sqrt[2]*a^(5/2)*f) - Sec[e + f*x]/(6*f*(a + a*Sin[e + f*x])^(5/2)) - (11*Cos[e + f*x])/(128*a*f*(a + a*Sin[e + f*x])^(3/2)) + (17*Sec[e + f*x])/(48*a*f*(a + a*Sin[e + f*x])^(3/2)) + (11*Sec[e + f*x])/(96*a^2*f*Sqrt[a + a*Sin[e + f*x]])","A",6,6,23,0.2609,1,"{2712, 2859, 2687, 2650, 2649, 206}"
113,1,141,0,0.346435,"\int \frac{\cot ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2),x]","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{2} a^{5/2} f}-\frac{2 \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{\cot (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{2} a^{5/2} f}-\frac{2 \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{\cot (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(5*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(a^(5/2)*f) - (7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) - (2*Cos[e + f*x])/(a*f*(a + a*Sin[e + f*x])^(3/2)) - Cot[e + f*x]/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",7,6,23,0.2609,1,"{2715, 2978, 2985, 2649, 206, 2773}"
114,1,191,0,0.9587343,"\int \frac{\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{19 \cot (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{45 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 a^{5/2} f}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt{a \sin (e+f x)+a}}","-\frac{19 \cot (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{45 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{8 a^{5/2} f}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt{a \sin (e+f x)+a}}",1,"(45*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*a^(5/2)*f) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*f) - (19*Cot[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (13*Cot[e + f*x]*Csc[e + f*x])/(12*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]])","A",16,8,23,0.3478,1,"{2717, 2779, 2984, 2985, 2649, 206, 2773, 3044}"
115,1,982,0,1.2705996,"\int \sqrt[3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(1/3)*Tan[e + f*x]^4,x]","-\frac{3 \sin ^2(e+f x) \tan (e+f x) a^2}{f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}+\frac{3 \sin (e+f x) \tan (e+f x) a^2}{2 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{\sec (e+f x) \left(65 a^2-142 a^2 \sin (e+f x)\right)}{42 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{361 \sec (e+f x) \sqrt[3]{\sin (e+f x) a+a}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt[3]{\sin (e+f x) a+a}}{63 f}+\frac{361 \left(1+\sqrt{3}\right) \sec (e+f x) (1-\sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}{63 f \left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)}-\frac{361 \sqrt[3]{2} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{\sin (e+f x) a+a} \sqrt[3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}}}{21\ 3^{3/4} f \sqrt{-\frac{\sqrt[3]{\sin (e+f x) a+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}} a^{2/3}}-\frac{361 \left(1-\sqrt{3}\right) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{\sin (e+f x) a+a} \sqrt[3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}}}{63\ 2^{2/3} \sqrt[4]{3} f \sqrt{-\frac{\sqrt[3]{\sin (e+f x) a+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}} a^{2/3}}","-\frac{3 \sin ^2(e+f x) \tan (e+f x) a^2}{f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}+\frac{3 \sin (e+f x) \tan (e+f x) a^2}{2 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{\sec (e+f x) \left(65 a^2-142 a^2 \sin (e+f x)\right)}{42 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{361 \sec (e+f x) \sqrt[3]{\sin (e+f x) a+a}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt[3]{\sin (e+f x) a+a}}{63 f}+\frac{361 \left(1+\sqrt{3}\right) \sec (e+f x) (1-\sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}{63 f \left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)}-\frac{361 \sqrt[3]{2} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{\sin (e+f x) a+a} \sqrt[3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}}}{21\ 3^{3/4} f \sqrt{-\frac{\sqrt[3]{\sin (e+f x) a+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}} a^{2/3}}-\frac{361 \left(1-\sqrt{3}\right) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{\sin (e+f x) a+a} \sqrt[3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}}}{63\ 2^{2/3} \sqrt[4]{3} f \sqrt{-\frac{\sqrt[3]{\sin (e+f x) a+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{\sin (e+f x) a+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}\right)^2}} a^{2/3}}",1,"(-361*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/(126*f) + (361*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(1/3))/(63*f) - (Sec[e + f*x]*(65*a^2 - 142*a^2*Sin[e + f*x]))/(42*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3)) + (361*(1 + Sqrt[3])*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))/(63*f*(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))) - (361*2^(1/3)*EllipticE[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (2 + Sqrt[3])/4]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(21*3^(3/4)*a^(2/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) - (361*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (2 + Sqrt[3])/4]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(63*2^(2/3)*3^(1/4)*a^(2/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(2*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3)) - (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))","A",10,9,23,0.3913,1,"{2719, 100, 153, 144, 51, 63, 308, 225, 1881}"
116,1,123,0,0.1933696,"\int \sqrt[3]{a+a \sin (e+f x)} \tan ^2(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^(1/3)*Tan[e + f*x]^2,x]","-\frac{5 a \sqrt[6]{\sin (e+f x)+1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 \sqrt[6]{2} f (a \sin (e+f x)+a)^{2/3}}-\frac{3 \sec (e+f x) (a \sin (e+f x)+a)^{4/3}}{a f}+\frac{7 \sec (e+f x) \sqrt[3]{a \sin (e+f x)+a}}{f}","-\frac{5 a \sqrt[6]{\sin (e+f x)+1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 \sqrt[6]{2} f (a \sin (e+f x)+a)^{2/3}}-\frac{3 \sec (e+f x) (a \sin (e+f x)+a)^{4/3}}{a f}+\frac{7 \sec (e+f x) \sqrt[3]{a \sin (e+f x)+a}}{f}",1,"(-5*a*Cos[e + f*x]*Hypergeometric2F1[1/2, 7/6, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(1/6))/(3*2^(1/6)*f*(a + a*Sin[e + f*x])^(2/3)) + (7*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/f - (3*Sec[e + f*x]*(a + a*Sin[e + f*x])^(4/3))/(a*f)","A",4,4,23,0.1739,1,"{2713, 2855, 2652, 2651}"
117,1,80,0,0.1013937,"\int \cot ^2(e+f x) \sqrt[3]{a+a \sin (e+f x)} \, dx","Int[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(1/3),x]","\frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{7/3} F_1\left(\frac{11}{6};-\frac{1}{2},2;\frac{17}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{11 a^2 f}","\frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{7/3} F_1\left(\frac{11}{6};-\frac{1}{2},2;\frac{17}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{11 a^2 f}",1,"(6*Sqrt[2]*AppellF1[11/6, -1/2, 2, 17/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(7/3))/(11*a^2*f)","A",3,3,23,0.1304,1,"{2719, 137, 136}"
118,1,80,0,0.0938648,"\int \cot ^4(e+f x) \sqrt[3]{a+a \sin (e+f x)} \, dx","Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(1/3),x]","\frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{10/3} F_1\left(\frac{17}{6};-\frac{3}{2},4;\frac{23}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{17 a^3 f}","\frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{10/3} F_1\left(\frac{17}{6};-\frac{3}{2},4;\frac{23}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{17 a^3 f}",1,"(12*Sqrt[2]*AppellF1[17/6, -3/2, 4, 23/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(10/3))/(17*a^3*f)","A",3,3,23,0.1304,1,"{2719, 137, 136}"
119,1,551,0,0.4935725,"\int \frac{\tan ^4(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Int[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]","\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a \sin (e+f x)+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{a \sin (e+f x)+a}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{495 \sqrt[3]{2} \sqrt[4]{3} a^{4/3} f \sqrt{-\frac{\sqrt[3]{a \sin (e+f x)+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a \sin (e+f x)+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{a \sin (e+f x)+a}\right)^2}}}-\frac{\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x)}{396 f \sqrt[3]{a \sin (e+f x)+a}}-\frac{973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt[3]{a \sin (e+f x)+a}}","\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a \sin (e+f x)+a}\right) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{a \sin (e+f x)+a}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2} \sqrt[3]{a}-\left(1-\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}{\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{\sin (e+f x) a+a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{495 \sqrt[3]{2} \sqrt[4]{3} a^{4/3} f \sqrt{-\frac{\sqrt[3]{a \sin (e+f x)+a} \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a \sin (e+f x)+a}\right)}{\left(\sqrt[3]{2} \sqrt[3]{a}-\left(1+\sqrt{3}\right) \sqrt[3]{a \sin (e+f x)+a}\right)^2}}}-\frac{\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x)}{396 f \sqrt[3]{a \sin (e+f x)+a}}-\frac{973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt[3]{a \sin (e+f x)+a}}",1,"(973*Sec[e + f*x])/(396*f*(a + a*Sin[e + f*x])^(1/3)) - (973*Sec[e + f*x]*(1 - Sin[e + f*x]))/(495*f*(a + a*Sin[e + f*x])^(1/3)) - (Sec[e + f*x]*(95*a + 356*a*Sin[e + f*x]))/(132*f*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3)) + (973*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (2 + Sqrt[3])/4]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(495*2^(1/3)*3^(1/4)*a^(4/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(4*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3)) + (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3))","A",8,7,23,0.3043,1,"{2719, 100, 153, 144, 51, 63, 225}"
120,1,126,0,0.2164936,"\int \frac{\tan ^2(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Int[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]","\frac{11 \sqrt[6]{2} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{15 f \sqrt[6]{\sin (e+f x)+1} \sqrt[3]{a \sin (e+f x)+a}}+\frac{4 \sec (e+f x) (a \sin (e+f x)+a)^{2/3}}{5 a f}-\frac{3 \sec (e+f x)}{5 f \sqrt[3]{a \sin (e+f x)+a}}","\frac{11 \sqrt[6]{2} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{15 f \sqrt[6]{\sin (e+f x)+1} \sqrt[3]{a \sin (e+f x)+a}}+\frac{4 \sec (e+f x) (a \sin (e+f x)+a)^{2/3}}{5 a f}-\frac{3 \sec (e+f x)}{5 f \sqrt[3]{a \sin (e+f x)+a}}",1,"(-3*Sec[e + f*x])/(5*f*(a + a*Sin[e + f*x])^(1/3)) + (11*2^(1/6)*Cos[e + f*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[e + f*x])/2])/(15*f*(1 + Sin[e + f*x])^(1/6)*(a + a*Sin[e + f*x])^(1/3)) + (4*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3))/(5*a*f)","A",4,4,23,0.1739,1,"{2712, 2855, 2652, 2651}"
121,1,80,0,0.0949979,"\int \frac{\cot ^2(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Int[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]","\frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left(\frac{7}{6};-\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{7 a^2 f}","\frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left(\frac{7}{6};-\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{7 a^2 f}",1,"(6*Sqrt[2]*AppellF1[7/6, -1/2, 2, 13/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(5/3))/(7*a^2*f)","A",3,3,23,0.1304,1,"{2719, 137, 136}"
122,1,80,0,0.0935749,"\int \frac{\cot ^4(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Int[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]","\frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{8/3} F_1\left(\frac{13}{6};-\frac{3}{2},4;\frac{19}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{13 a^3 f}","\frac{12 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{8/3} F_1\left(\frac{13}{6};-\frac{3}{2},4;\frac{19}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{13 a^3 f}",1,"(12*Sqrt[2]*AppellF1[13/6, -3/2, 4, 19/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(8/3))/(13*a^3*f)","A",3,3,23,0.1304,1,"{2719, 137, 136}"
123,1,269,0,0.3451075,"\int (a+a \sin (e+f x))^3 (g \tan (e+f x))^p \, dx","Int[(a + a*Sin[e + f*x])^3*(g*Tan[e + f*x])^p,x]","\frac{3 a^3 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{a^3 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{a^3 \sin ^3(e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+4}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{f g (p+4)}+\frac{3 a^3 \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}","\frac{3 a^3 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{a^3 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{a^3 \sin ^3(e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+4}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{f g (p+4)}+\frac{3 a^3 \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}",1,"(a^3*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (3*a^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p)) + (a^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (4 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*(g*Tan[e + f*x])^(1 + p))/(f*g*(4 + p)) + (3*a^3*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))","A",10,6,23,0.2609,1,"{2710, 3476, 364, 2602, 2577, 2591}"
124,1,187,0,0.2265235,"\int (a+a \sin (e+f x))^2 (g \tan (e+f x))^p \, dx","Int[(a + a*Sin[e + f*x])^2*(g*Tan[e + f*x])^p,x]","\frac{a^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{a^2 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{2 a^2 \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}","\frac{a^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{a^2 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{2 a^2 \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}",1,"(a^2*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (2*a^2*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p)) + (a^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))","A",8,6,23,0.2609,1,"{2710, 3476, 364, 2602, 2577, 2591}"
125,1,129,0,0.1387814,"\int (a+a \sin (e+f x)) (g \tan (e+f x))^p \, dx","Int[(a + a*Sin[e + f*x])*(g*Tan[e + f*x])^p,x]","\frac{a (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{a \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}","\frac{a (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{a \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}",1,"(a*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (a*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p))","A",6,5,21,0.2381,1,"{2710, 3476, 364, 2602, 2577}"
126,1,108,0,0.1261203,"\int \frac{(g \tan (e+f x))^p}{a+a \sin (e+f x)} \, dx","Int[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x]),x]","\frac{(g \tan (e+f x))^{p+1}}{a f g (p+1)}-\frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{p+3}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+3}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a f g^2 (p+2)}","\frac{(g \tan (e+f x))^{p+1}}{a f g (p+1)}-\frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{p+3}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+3}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a f g^2 (p+2)}",1,"(g*Tan[e + f*x])^(1 + p)/(a*f*g*(1 + p)) - ((Cos[e + f*x]^2)^((3 + p)/2)*Hypergeometric2F1[(2 + p)/2, (3 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(g*Tan[e + f*x])^(2 + p))/(a*f*g^2*(2 + p))","A",4,4,23,0.1739,1,"{2706, 2607, 32, 2617}"
127,1,138,0,0.2747544,"\int \frac{(g \tan (e+f x))^p}{(a+a \sin (e+f x))^2} \, dx","Int[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^2,x]","-\frac{2 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{p+5}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+5}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a^2 f g^2 (p+2)}+\frac{2 (g \tan (e+f x))^{p+3}}{a^2 f g^3 (p+3)}+\frac{(g \tan (e+f x))^{p+1}}{a^2 f g (p+1)}","-\frac{2 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{p+5}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+5}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a^2 f g^2 (p+2)}+\frac{2 (g \tan (e+f x))^{p+3}}{a^2 f g^3 (p+3)}+\frac{(g \tan (e+f x))^{p+1}}{a^2 f g (p+1)}",1,"(g*Tan[e + f*x])^(1 + p)/(a^2*f*g*(1 + p)) - (2*(Cos[e + f*x]^2)^((5 + p)/2)*Hypergeometric2F1[(2 + p)/2, (5 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(g*Tan[e + f*x])^(2 + p))/(a^2*f*g^2*(2 + p)) + (2*(g*Tan[e + f*x])^(3 + p))/(a^2*f*g^3*(3 + p))","A",10,6,23,0.2609,1,"{2711, 2607, 14, 16, 2617, 32}"
128,1,248,0,0.4218134,"\int \frac{(g \tan (e+f x))^p}{(a+a \sin (e+f x))^3} \, dx","Int[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^3,x]","-\frac{3 \sec ^5(e+f x) \cos ^2(e+f x)^{\frac{p+7}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+7}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a^3 f g^2 (p+2)}-\frac{\sec ^3(e+f x) \cos ^2(e+f x)^{\frac{p+7}{2}} (g \tan (e+f x))^{p+4} \, _2F_1\left(\frac{p+4}{2},\frac{p+7}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{a^3 f g^4 (p+4)}+\frac{5 (g \tan (e+f x))^{p+3}}{a^3 f g^3 (p+3)}+\frac{4 (g \tan (e+f x))^{p+5}}{a^3 f g^5 (p+5)}+\frac{(g \tan (e+f x))^{p+1}}{a^3 f g (p+1)}","-\frac{3 \sec ^5(e+f x) \cos ^2(e+f x)^{\frac{p+7}{2}} (g \tan (e+f x))^{p+2} \, _2F_1\left(\frac{p+2}{2},\frac{p+7}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{a^3 f g^2 (p+2)}-\frac{\sec ^3(e+f x) \cos ^2(e+f x)^{\frac{p+7}{2}} (g \tan (e+f x))^{p+4} \, _2F_1\left(\frac{p+4}{2},\frac{p+7}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{a^3 f g^4 (p+4)}+\frac{5 (g \tan (e+f x))^{p+3}}{a^3 f g^3 (p+3)}+\frac{4 (g \tan (e+f x))^{p+5}}{a^3 f g^5 (p+5)}+\frac{(g \tan (e+f x))^{p+1}}{a^3 f g (p+1)}",1,"(g*Tan[e + f*x])^(1 + p)/(a^3*f*g*(1 + p)) - (3*(Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(2 + p)/2, (7 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^5*(g*Tan[e + f*x])^(2 + p))/(a^3*f*g^2*(2 + p)) + (5*(g*Tan[e + f*x])^(3 + p))/(a^3*f*g^3*(3 + p)) - ((Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(4 + p)/2, (7 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(g*Tan[e + f*x])^(4 + p))/(a^3*f*g^4*(4 + p)) + (4*(g*Tan[e + f*x])^(5 + p))/(a^3*f*g^5*(5 + p))","A",13,6,23,0.2609,1,"{2711, 2607, 270, 16, 2617, 14}"
129,1,111,0,0.1240095,"\int (a+a \sin (e+f x))^m (g \tan (e+f x))^p \, dx","Int[(a + a*Sin[e + f*x])^m*(g*Tan[e + f*x])^p,x]","\frac{(1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^m (g \tan (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac{1}{2} (-2 m+p+1)} F_1\left(p+1;\frac{p+1}{2},\frac{1}{2} (-2 m+p+1);p+2;\sin (e+f x),-\sin (e+f x)\right)}{f g (p+1)}","\frac{(1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^m (g \tan (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac{1}{2} (-2 m+p+1)} F_1\left(p+1;\frac{p+1}{2},\frac{1}{2} (-2 m+p+1);p+2;\sin (e+f x),-\sin (e+f x)\right)}{f g (p+1)}",1,"(AppellF1[1 + p, (1 + p)/2, (1 - 2*m + p)/2, 2 + p, Sin[e + f*x], -Sin[e + f*x]]*(1 - Sin[e + f*x])^((1 + p)/2)*(1 + Sin[e + f*x])^((1 - 2*m + p)/2)*(a + a*Sin[e + f*x])^m*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p))","A",4,3,23,0.1304,1,"{2720, 135, 133}"
130,1,163,0,0.1487529,"\int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^3,x]","-\frac{a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{f m (a-a \sin (e+f x))}+\frac{a (m+4) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 f (1-m)}+\frac{\left(2 a m \sin (e+f x)+a \left(-m^2-3 m+2\right)\right) (a \sin (e+f x)+a)^{m-1}}{2 f (1-m) m (1-\sin (e+f x))}","-\frac{a^2 \sin ^2(e+f x) (a \sin (e+f x)+a)^{m-1}}{f m (a-a \sin (e+f x))}+\frac{a (m+4) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 f (1-m)}+\frac{\left(2 a m \sin (e+f x)+a \left(-m^2-3 m+2\right)\right) (a \sin (e+f x)+a)^{m-1}}{2 f (1-m) m (1-\sin (e+f x))}",1,"(a*(4 + m)*Hypergeometric2F1[1, -1 + m, m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(-1 + m))/(4*f*(1 - m)) - (a^2*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-1 + m))/(f*m*(a - a*Sin[e + f*x])) + ((a + a*Sin[e + f*x])^(-1 + m)*(a*(2 - 3*m - m^2) + 2*a*m*Sin[e + f*x]))/(2*f*(1 - m)*m*(1 - Sin[e + f*x]))","A",4,4,21,0.1905,1,"{2707, 100, 146, 68}"
131,1,72,0,0.0504618,"\int (a+a \sin (e+f x))^m \tan (e+f x) \, dx","Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x],x]","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 a f (m+1)}-\frac{(a \sin (e+f x)+a)^m}{2 f m}","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 a f (m+1)}-\frac{(a \sin (e+f x)+a)^m}{2 f m}",1,"-(a + a*Sin[e + f*x])^m/(2*f*m) + (Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(1 + m))","A",3,3,19,0.1579,1,"{2707, 79, 68}"
132,1,43,0,0.0439523,"\int \cot (e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cot[e + f*x]*(a + a*Sin[e + f*x])^m,x]","-\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)}","-\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)}",1,"-((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)))","A",2,2,19,0.1053,1,"{2707, 65}"
133,1,83,0,0.0685841,"\int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cot[e + f*x]^3*(a + a*Sin[e + f*x])^m,x]","-\frac{(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac{\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f}","-\frac{(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac{\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f}",1,"-(Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(2 + m))/(2*a^2*f) - ((2 - m)*Hypergeometric2F1[2, 2 + m, 3 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(2 + m))/(2*a^2*f*(2 + m))","A",3,3,21,0.1429,1,"{2707, 78, 65}"
134,1,123,0,0.0982222,"\int \cot ^5(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cot[e + f*x]^5*(a + a*Sin[e + f*x])^m,x]","-\frac{\left(m^2-9 m+12\right) (a \sin (e+f x)+a)^{m+3} \, _2F_1(3,m+3;m+4;\sin (e+f x)+1)}{12 a^3 f (m+3)}-\frac{\csc ^4(e+f x) (a \sin (e+f x)+a)^{m+3}}{4 a^3 f}+\frac{(9-m) \csc ^3(e+f x) (a \sin (e+f x)+a)^{m+3}}{12 a^3 f}","-\frac{\left(m^2-9 m+12\right) (a \sin (e+f x)+a)^{m+3} \, _2F_1(3,m+3;m+4;\sin (e+f x)+1)}{12 a^3 f (m+3)}-\frac{\csc ^4(e+f x) (a \sin (e+f x)+a)^{m+3}}{4 a^3 f}+\frac{(9-m) \csc ^3(e+f x) (a \sin (e+f x)+a)^{m+3}}{12 a^3 f}",1,"((9 - m)*Csc[e + f*x]^3*(a + a*Sin[e + f*x])^(3 + m))/(12*a^3*f) - (Csc[e + f*x]^4*(a + a*Sin[e + f*x])^(3 + m))/(4*a^3*f) - ((12 - 9*m + m^2)*Hypergeometric2F1[3, 3 + m, 4 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(3 + m))/(12*a^3*f*(3 + m))","A",4,4,21,0.1905,1,"{2707, 89, 78, 65}"
135,1,311,0,0.3550584,"\int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]","-\frac{a^2 \sin ^2(e+f x) \tan (e+f x) (a \sin (e+f x)+a)^{m-1}}{f m (a-a \sin (e+f x))}+\frac{a^2 \sin (e+f x) \tan (e+f x) (a \sin (e+f x)+a)^{m-1}}{f (1-m) (a-a \sin (e+f x))}+\frac{2^{m-\frac{3}{2}} \left(m^4+6 m^3-7 m^2-12 m+9\right) (1-\sin (e+f x)) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{5}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f (1-m) m}-\frac{\sec (e+f x) (a \sin (e+f x)+a)^{m-1} \left(a \left(-m^3-7 m^2-m+6\right)-a \left(-m^3-8 m^2-6 m+9\right) \sin (e+f x)\right)}{3 f (1-m) m (1-\sin (e+f x))}","-\frac{a^2 \sin ^2(e+f x) \tan (e+f x) (a \sin (e+f x)+a)^{m-1}}{f m (a-a \sin (e+f x))}+\frac{a^2 \sin (e+f x) \tan (e+f x) (a \sin (e+f x)+a)^{m-1}}{f (1-m) (a-a \sin (e+f x))}+\frac{2^{m-\frac{3}{2}} \left(m^4+6 m^3-7 m^2-12 m+9\right) (1-\sin (e+f x)) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{5}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f (1-m) m}-\frac{\sec (e+f x) (a \sin (e+f x)+a)^{m-1} \left(a \left(-m^3-7 m^2-m+6\right)-a \left(-m^3-8 m^2-6 m+9\right) \sin (e+f x)\right)}{3 f (1-m) m (1-\sin (e+f x))}",1,"(2^(-3/2 + m)*(9 - 12*m - 7*m^2 + 6*m^3 + m^4)*Hypergeometric2F1[1/2, 5/2 - m, 3/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(3*f*(1 - m)*m) - (Sec[e + f*x]*(a + a*Sin[e + f*x])^(-1 + m)*(a*(6 - m - 7*m^2 - m^3) - a*(9 - 6*m - 8*m^2 - m^3)*Sin[e + f*x]))/(3*f*(1 - m)*m*(1 - Sin[e + f*x])) + (a^2*Sin[e + f*x]*(a + a*Sin[e + f*x])^(-1 + m)*Tan[e + f*x])/(f*(1 - m)*(a - a*Sin[e + f*x])) - (a^2*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-1 + m)*Tan[e + f*x])/(f*m*(a - a*Sin[e + f*x]))","A",6,6,21,0.2857,1,"{2719, 100, 153, 145, 70, 69}"
136,1,157,0,0.2480164,"\int (a+a \sin (e+f x))^m \tan ^2(e+f x) \, dx","Int[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^2,x]","\frac{2^{m-\frac{1}{2}} \left(-m^2-m+1\right) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (1-m) m}+\frac{\sec (e+f x) (a \sin (e+f x)+a)^m}{f (1-m) m}-\frac{\sec (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f m}","\frac{2^{m-\frac{1}{2}} \left(-m^2-m+1\right) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (1-m) m}+\frac{\sec (e+f x) (a \sin (e+f x)+a)^m}{f (1-m) m}-\frac{\sec (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f m}",1,"(Sec[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 - m)*m) + (2^(-1/2 + m)*(1 - m - m^2)*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 - m)*m) - (Sec[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*m)","A",5,5,21,0.2381,1,"{2713, 2860, 2689, 70, 69}"
137,1,74,0,0.0348217,"\int (a+a \sin (e+f x))^m \, dx","Int[(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-((2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)","A",2,2,12,0.1667,1,"{2652, 2651}"
138,1,89,0,0.10163,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^m,x]","\frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+2} F_1\left(m+\frac{3}{2};-\frac{1}{2},2;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{a^2 f (2 m+3)}","\frac{2 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+2} F_1\left(m+\frac{3}{2};-\frac{1}{2},2;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{a^2 f (2 m+3)}",1,"(2*Sqrt[2]*AppellF1[3/2 + m, -1/2, 2, 5/2 + m, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(3 + 2*m))","A",3,3,21,0.1429,1,"{2719, 137, 136}"
139,1,89,0,0.0993711,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^m,x]","\frac{4 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left(m+\frac{5}{2};-\frac{3}{2},4;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{a^3 f (2 m+5)}","\frac{4 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left(m+\frac{5}{2};-\frac{3}{2},4;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right)}{a^3 f (2 m+5)}",1,"(4*Sqrt[2]*AppellF1[5/2 + m, -3/2, 4, 7/2 + m, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(5 + 2*m))","A",3,3,21,0.1429,1,"{2719, 137, 136}"
140,1,88,0,0.0771489,"\int (a+b \sin (c+d x)) \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{(2 a+3 b) \log (1-\sin (c+d x))}{4 d}+\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d}+\frac{\tan ^2(c+d x) (a+b \sin (c+d x))}{2 d}+\frac{3 b \sin (c+d x)}{2 d}","\frac{(2 a+3 b) \log (1-\sin (c+d x))}{4 d}+\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d}+\frac{\tan ^2(c+d x) (a+b \sin (c+d x))}{2 d}+\frac{3 b \sin (c+d x)}{2 d}",1,"((2*a + 3*b)*Log[1 - Sin[c + d*x]])/(4*d) + ((2*a - 3*b)*Log[1 + Sin[c + d*x]])/(4*d) + (3*b*Sin[c + d*x])/(2*d) + ((a + b*Sin[c + d*x])*Tan[c + d*x]^2)/(2*d)","A",6,5,19,0.2632,1,"{2721, 819, 774, 633, 31}"
141,1,55,0,0.0378478,"\int (a+b \sin (c+d x)) \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x],x]","-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}-\frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{b \sin (c+d x)}{d}","-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}-\frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{b \sin (c+d x)}{d}",1,"-((a + b)*Log[1 - Sin[c + d*x]])/(2*d) - ((a - b)*Log[1 + Sin[c + d*x]])/(2*d) - (b*Sin[c + d*x])/d","A",5,4,17,0.2353,1,"{2721, 774, 633, 31}"
142,1,24,0,0.0213609,"\int \cot (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}","\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}",1,"(a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d","A",3,2,17,0.1176,1,"{2721, 43}"
143,1,54,0,0.0409958,"\int \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \log (\sin (c+d x))}{d}-\frac{b \sin (c+d x)}{d}-\frac{b \csc (c+d x)}{d}","-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \log (\sin (c+d x))}{d}-\frac{b \sin (c+d x)}{d}-\frac{b \csc (c+d x)}{d}",1,"-((b*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2721, 766}"
144,1,81,0,0.0521741,"\int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d}","-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d}",1,"(2*b*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (b*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2721, 766}"
145,1,72,0,0.0778072,"\int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+a x-\frac{b \cos (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d}-\frac{2 b \sec (c+d x)}{d}","\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+a x-\frac{b \cos (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d}-\frac{2 b \sec (c+d x)}{d}",1,"a*x - (b*Cos[c + d*x])/d - (2*b*Sec[c + d*x])/d + (b*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",8,5,19,0.2632,1,"{2722, 3473, 8, 2590, 270}"
146,1,38,0,0.0593762,"\int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}",1,"-(a*x) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",7,5,19,0.2632,1,"{2722, 3473, 8, 2590, 14}"
147,1,41,0,0.0533796,"\int \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{a \cot (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}",1,"-(a*x) - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d - (a*Cot[c + d*x])/d","A",7,6,19,0.3158,1,"{2722, 2592, 321, 206, 3473, 8}"
148,1,82,0,0.0798351,"\int \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}+a x-\frac{3 b \cos (c+d x)}{2 d}-\frac{b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}+a x-\frac{3 b \cos (c+d x)}{2 d}-\frac{b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{2 d}",1,"a*x + (3*b*ArcTanh[Cos[c + d*x]])/(2*d) - (3*b*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)","A",9,7,19,0.3684,1,"{2722, 2592, 288, 321, 206, 3473, 8}"
149,1,122,0,0.0972308,"\int \cot ^6(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-a x+\frac{15 b \cos (c+d x)}{8 d}-\frac{b \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 b \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 b \tanh ^{-1}(\cos (c+d x))}{8 d}","-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-a x+\frac{15 b \cos (c+d x)}{8 d}-\frac{b \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 b \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 b \tanh ^{-1}(\cos (c+d x))}{8 d}",1,"-(a*x) - (15*b*ArcTanh[Cos[c + d*x]])/(8*d) + (15*b*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*b*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)","A",11,7,19,0.3684,1,"{2722, 2592, 288, 321, 206, 3473, 8}"
150,1,111,0,0.1729345,"\int (a+b \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{2 a b \sin (c+d x)}{d}+\frac{(a+b) (a+2 b) \log (1-\sin (c+d x))}{2 d}+\frac{(a-2 b) (a-b) \log (\sin (c+d x)+1)}{2 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{b^2 \sin ^2(c+d x)}{2 d}","\frac{2 a b \sin (c+d x)}{d}+\frac{(a+b) (a+2 b) \log (1-\sin (c+d x))}{2 d}+\frac{(a-2 b) (a-b) \log (\sin (c+d x)+1)}{2 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"((a + b)*(a + 2*b)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - 2*b)*(a - b)*Log[1 + Sin[c + d*x]])/(2*d) + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(2*d)","A",7,5,21,0.2381,1,"{2721, 1645, 1629, 633, 31}"
151,1,78,0,0.0709252,"\int (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{2 a b \sin (c+d x)}{d}-\frac{(a-b)^2 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 \log (1-\sin (c+d x))}{2 d}-\frac{b^2 \sin ^2(c+d x)}{2 d}","-\frac{2 a b \sin (c+d x)}{d}-\frac{(a-b)^2 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 \log (1-\sin (c+d x))}{2 d}-\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"-((a + b)^2*Log[1 - Sin[c + d*x]])/(2*d) - ((a - b)^2*Log[1 + Sin[c + d*x]])/(2*d) - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d)","A",6,4,19,0.2105,1,"{2721, 801, 633, 31}"
152,1,46,0,0.038808,"\int \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}","\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"(a^2*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{2721, 43}"
153,1,84,0,0.0730727,"\int \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \sin ^2(c+d x)}{2 d}","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"(-2*a*b*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{2721, 894}"
154,1,126,0,0.1035415,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 d}+\frac{\left(a^2-2 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{4 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 d}+\frac{\left(a^2-2 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{4 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"(4*a*b*Csc[c + d*x])/d + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) + ((a^2 - 2*b^2)*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{2721, 948}"
155,1,149,0,0.161982,"\int (a+b \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x-\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{4 a b \sec (c+d x)}{d}+\frac{5 b^2 \tan ^3(c+d x)}{6 d}-\frac{5 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}","\frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x-\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{4 a b \sec (c+d x)}{d}+\frac{5 b^2 \tan ^3(c+d x)}{6 d}-\frac{5 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}",1,"a^2*x + (5*b^2*x)/2 - (2*a*b*Cos[c + d*x])/d - (4*a*b*Sec[c + d*x])/d + (2*a*b*Sec[c + d*x]^3)/(3*d) - (a^2*Tan[c + d*x])/d - (5*b^2*Tan[c + d*x])/(2*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (5*b^2*Tan[c + d*x]^3)/(6*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)","A",13,9,21,0.4286,1,"{2722, 3473, 8, 2590, 270, 2591, 288, 302, 203}"
156,1,94,0,0.1223154,"\int (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2}","\frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2}",1,"-(a^2*x) - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c + d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",11,9,21,0.4286,1,"{2722, 3473, 8, 2590, 14, 2591, 288, 321, 203}"
157,1,78,0,0.0855624,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b^2 x}{2}","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b^2 x}{2}",1,"-(a^2*x) + (b^2*x)/2 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",9,7,21,0.3333,1,"{2722, 2635, 8, 2592, 321, 206, 3473}"
158,1,133,0,0.1493955,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{3 a b \cos (c+d x)}{d}-\frac{a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b^2 x}{2}","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{3 a b \cos (c+d x)}{d}-\frac{a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b^2 x}{2}",1,"a^2*x - (3*b^2*x)/2 + (3*a*b*ArcTanh[Cos[c + d*x]])/d - (3*a*b*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (3*b^2*Cot[c + d*x])/(2*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d)","A",13,9,21,0.4286,1,"{2722, 2591, 288, 321, 203, 2592, 206, 3473, 8}"
159,1,202,0,0.1694846,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}",1,"-(a^2*x) + (5*b^2*x)/2 - (15*a*b*ArcTanh[Cos[c + d*x]])/(4*d) + (15*a*b*Cos[c + d*x])/(4*d) - (a^2*Cot[c + d*x])/d + (5*b^2*Cot[c + d*x])/(2*d) + (5*a*b*Cos[c + d*x]*Cot[c + d*x]^2)/(4*d) + (a^2*Cot[c + d*x]^3)/(3*d) - (5*b^2*Cot[c + d*x]^3)/(6*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d)","A",16,10,21,0.4762,1,"{2722, 2591, 288, 302, 203, 2592, 321, 206, 3473, 8}"
160,1,150,0,0.241513,"\int (a+b \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{b \left(6 a^2+5 b^2\right) \sin (c+d x)}{2 d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{(a+b)^2 (2 a+5 b) \log (1-\sin (c+d x))}{4 d}+\frac{(2 a-5 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}","\frac{b \left(6 a^2+5 b^2\right) \sin (c+d x)}{2 d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{(a+b)^2 (2 a+5 b) \log (1-\sin (c+d x))}{4 d}+\frac{(2 a-5 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"((a + b)^2*(2*a + 5*b)*Log[1 - Sin[c + d*x]])/(4*d) + ((2*a - 5*b)*(a - b)^2*Log[1 + Sin[c + d*x]])/(4*d) + (b*(6*a^2 + 5*b^2)*Sin[c + d*x])/(2*d) + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(2*d)","A",7,5,21,0.2381,1,"{2721, 1645, 1629, 633, 31}"
161,1,105,0,0.1135935,"\int (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{b \left(3 a^2+b^2\right) \sin (c+d x)}{d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}-\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}","-\frac{b \left(3 a^2+b^2\right) \sin (c+d x)}{d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}-\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"-((a + b)^3*Log[1 - Sin[c + d*x]])/(2*d) - ((a - b)^3*Log[1 + Sin[c + d*x]])/(2*d) - (b*(3*a^2 + b^2)*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d)","A",6,4,19,0.2105,1,"{2721, 801, 633, 31}"
162,1,67,0,0.0449595,"\int \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}","\frac{3 a^2 b \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(a^3*Log[Sin[c + d*x]])/d + (3*a^2*b*Sin[c + d*x])/d + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2721, 43}"
163,1,116,0,0.0943571,"\int \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(3 a^2-b^2\right) \sin (c+d x)}{d}-\frac{a \left(a^2-3 b^2\right) \log (\sin (c+d x))}{d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}","-\frac{b \left(3 a^2-b^2\right) \sin (c+d x)}{d}-\frac{a \left(a^2-3 b^2\right) \log (\sin (c+d x))}{d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(-3*a^2*b*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (b*(3*a^2 - b^2)*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{2721, 894}"
164,1,165,0,0.1410297,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{b \left(3 a^2-2 b^2\right) \sin (c+d x)}{d}+\frac{a \left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 d}+\frac{b \left(6 a^2-b^2\right) \csc (c+d x)}{d}+\frac{a \left(a^2-6 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}","\frac{b \left(3 a^2-2 b^2\right) \sin (c+d x)}{d}+\frac{a \left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 d}+\frac{b \left(6 a^2-b^2\right) \csc (c+d x)}{d}+\frac{a \left(a^2-6 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(b*(6*a^2 - b^2)*Csc[c + d*x])/d + (a*(2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (a^3*Csc[c + d*x]^4)/(4*d) + (a*(a^2 - 6*b^2)*Log[Sin[c + d*x]])/d + (b*(3*a^2 - 2*b^2)*Sin[c + d*x])/d + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{2721, 948}"
165,1,220,0,0.2242994,"\int (a+b \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","-\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^2 b \sec ^3(c+d x)}{d}-\frac{6 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+a^3 x+\frac{5 a b^2 \tan ^3(c+d x)}{2 d}-\frac{15 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{15}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}-\frac{3 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec ^3(c+d x)}{3 d}-\frac{3 b^3 \sec (c+d x)}{d}","-\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^2 b \sec ^3(c+d x)}{d}-\frac{6 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+a^3 x+\frac{5 a b^2 \tan ^3(c+d x)}{2 d}-\frac{15 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{15}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}-\frac{3 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec ^3(c+d x)}{3 d}-\frac{3 b^3 \sec (c+d x)}{d}",1,"a^3*x + (15*a*b^2*x)/2 - (3*a^2*b*Cos[c + d*x])/d - (3*b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) - (6*a^2*b*Sec[c + d*x])/d - (3*b^3*Sec[c + d*x])/d + (a^2*b*Sec[c + d*x]^3)/d + (b^3*Sec[c + d*x]^3)/(3*d) - (a^3*Tan[c + d*x])/d - (15*a*b^2*Tan[c + d*x])/(2*d) + (a^3*Tan[c + d*x]^3)/(3*d) + (5*a*b^2*Tan[c + d*x]^3)/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)","A",16,9,21,0.4286,1,"{2722, 3473, 8, 2590, 270, 2591, 288, 302, 203}"
166,1,146,0,0.169452,"\int (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{2 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}","\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{2 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"-(a^3*x) - (9*a*b^2*x)/2 + (3*a^2*b*Cos[c + d*x])/d + (2*b^3*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (9*a*b^2*Tan[c + d*x])/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",14,10,21,0.4762,1,"{2722, 3473, 8, 2590, 14, 2591, 288, 321, 203, 270}"
167,1,102,0,0.1118229,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}","\frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}",1,"-(a^3*x) + (3*a*b^2*x)/2 - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",11,9,21,0.4286,1,"{2722, 2635, 8, 2592, 321, 206, 3473, 2565, 30}"
168,1,194,0,0.1839182,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","-\frac{9 a^2 b \cos (c+d x)}{2 d}-\frac{3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}+a^3 x-\frac{9 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{9}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{b^3 \cos (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{9 a^2 b \cos (c+d x)}{2 d}-\frac{3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}+a^3 x-\frac{9 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{9}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{b^3 \cos (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"a^3*x - (9*a*b^2*x)/2 + (9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - (9*a^2*b*Cos[c + d*x])/(2*d) + (b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d)","A",17,10,21,0.4762,1,"{2722, 2592, 302, 206, 2591, 288, 321, 203, 3473, 8}"
169,1,291,0,0.2309,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]","\frac{45 a^2 b \cos (c+d x)}{8 d}-\frac{3 a^2 b \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{15 a^2 b \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{45 a^2 b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-a^3 x-\frac{5 a b^2 \cot ^3(c+d x)}{2 d}+\frac{15 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{15}{2} a b^2 x-\frac{5 b^3 \cos ^3(c+d x)}{6 d}-\frac{5 b^3 \cos (c+d x)}{2 d}-\frac{b^3 \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{5 b^3 \tanh ^{-1}(\cos (c+d x))}{2 d}","\frac{45 a^2 b \cos (c+d x)}{8 d}-\frac{3 a^2 b \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{15 a^2 b \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{45 a^2 b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-a^3 x-\frac{5 a b^2 \cot ^3(c+d x)}{2 d}+\frac{15 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{15}{2} a b^2 x-\frac{5 b^3 \cos ^3(c+d x)}{6 d}-\frac{5 b^3 \cos (c+d x)}{2 d}-\frac{b^3 \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{5 b^3 \tanh ^{-1}(\cos (c+d x))}{2 d}",1,"-(a^3*x) + (15*a*b^2*x)/2 - (45*a^2*b*ArcTanh[Cos[c + d*x]])/(8*d) + (5*b^3*ArcTanh[Cos[c + d*x]])/(2*d) + (45*a^2*b*Cos[c + d*x])/(8*d) - (5*b^3*Cos[c + d*x])/(2*d) - (5*b^3*Cos[c + d*x]^3)/(6*d) - (a^3*Cot[c + d*x])/d + (15*a*b^2*Cot[c + d*x])/(2*d) + (15*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) - (b^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d) + (a^3*Cot[c + d*x]^3)/(3*d) - (5*a*b^2*Cot[c + d*x]^3)/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d)","A",21,10,21,0.4762,1,"{2722, 2592, 288, 302, 206, 2591, 203, 321, 3473, 8}"
170,1,204,0,0.3630192,"\int \frac{\tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{a^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}","\frac{a^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}",1,"-((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(2*a^2 - b^2) - b*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",5,3,21,0.1429,1,"{2721, 1647, 801}"
171,1,126,0,0.1909221,"\int \frac{\tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}","-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",4,3,21,0.1429,1,"{2721, 1647, 801}"
172,1,74,0,0.0661064,"\int \frac{\tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{a \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}","\frac{a \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)","A",3,2,19,0.1053,1,"{2721, 801}"
173,1,34,0,0.0397367,"\int \frac{\cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (a+b \sin (c+d x))}{a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (a+b \sin (c+d x))}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{2721, 36, 29, 31}"
174,1,84,0,0.0888076,"\int \frac{\cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^3*d)","A",3,2,21,0.09524,1,"{2721, 894}"
175,1,148,0,0.1383524,"\int \frac{\cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^5 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{4 a d}","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^5 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"-((b*(2*a^2 - b^2)*Csc[c + d*x])/(a^4*d)) + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^3*d) + (b*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^5*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^5*d)","A",3,2,21,0.09524,1,"{2721, 894}"
176,1,177,0,0.2390692,"\int \frac{\tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 b \sec (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 b \sec (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}",1,"(2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (a^2*b*Sec[c + d*x])/((a^2 - b^2)^2*d) + (b*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^3*Tan[c + d*x])/((a^2 - b^2)^2*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",13,9,21,0.4286,1,"{2727, 2607, 30, 2606, 3767, 8, 2660, 618, 204}"
177,1,96,0,0.0988934,"\int \frac{\tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}","-\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}",1,"(-2*a^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)","A",8,7,21,0.3333,1,"{2727, 3767, 8, 2606, 2660, 618, 204}"
178,1,80,0,0.2427568,"\int \frac{\cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"(-2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d)","A",7,7,21,0.3333,1,"{2723, 3056, 3001, 3770, 2660, 618, 204}"
179,1,154,0,0.425707,"\int \frac{\cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*d) - (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)","A",7,7,21,0.3333,1,"{2725, 3055, 3001, 3770, 2660, 618, 204}"
180,1,307,0,1.106013,"\int \frac{\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-20 a^2 b^2+15 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(-22 a^2 b^2+15 a^4+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac{\left(-9 a^2 b^2+8 a^4+4 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-20 a^2 b^2+15 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(-22 a^2 b^2+15 a^4+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac{\left(-9 a^2 b^2+8 a^4+4 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}",1,"(-2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)","A",9,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
181,1,242,0,0.6326206,"\int \frac{\tan ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","-\frac{a^5}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{a^4 \left(a^2+5 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}+\frac{\sec ^4(c+d x) \left(a^2-2 a b \sin (c+d x)+b^2\right)}{4 d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) \left(2 \left(3 a^2 b^2+2 a^4-b^4\right)-a b \left(9 a^2-b^2\right) \sin (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}-\frac{a (4 a+b) \log (1-\sin (c+d x))}{8 d (a+b)^4}-\frac{a (4 a-b) \log (\sin (c+d x)+1)}{8 d (a-b)^4}","-\frac{a^5}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{a^4 \left(a^2+5 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}+\frac{\sec ^4(c+d x) \left(a^2-2 a b \sin (c+d x)+b^2\right)}{4 d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) \left(2 \left(3 a^2 b^2+2 a^4-b^4\right)-a b \left(9 a^2-b^2\right) \sin (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}-\frac{a (4 a+b) \log (1-\sin (c+d x))}{8 d (a+b)^4}-\frac{a (4 a-b) \log (\sin (c+d x)+1)}{8 d (a-b)^4}",1,"-(a*(4*a + b)*Log[1 - Sin[c + d*x]])/(8*(a + b)^4*d) - (a*(4*a - b)*Log[1 + Sin[c + d*x]])/(8*(a - b)^4*d) + (a^4*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - a^5/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(a^2 + b^2 - 2*a*b*Sin[c + d*x]))/(4*(a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(2*(2*a^4 + 3*a^2*b^2 - b^4) - a*b*(9*a^2 - b^2)*Sin[c + d*x]))/(4*(a^2 - b^2)^3*d)","A",5,3,21,0.1429,1,"{2721, 1647, 1629}"
182,1,161,0,0.3116955,"\int \frac{\tan ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","\frac{a^3}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{a^2 \left(a^2+3 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^2(c+d x) \left(a^2-2 a b \sin (c+d x)+b^2\right)}{2 d \left(a^2-b^2\right)^2}+\frac{a \log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac{a \log (\sin (c+d x)+1)}{2 d (a-b)^3}","\frac{a^3}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{a^2 \left(a^2+3 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^2(c+d x) \left(a^2-2 a b \sin (c+d x)+b^2\right)}{2 d \left(a^2-b^2\right)^2}+\frac{a \log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac{a \log (\sin (c+d x)+1)}{2 d (a-b)^3}",1,"(a*Log[1 - Sin[c + d*x]])/(2*(a + b)^3*d) + (a*Log[1 + Sin[c + d*x]])/(2*(a - b)^3*d) - (a^2*(a^2 + 3*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + a^3/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(a^2 + b^2 - 2*a*b*Sin[c + d*x]))/(2*(a^2 - b^2)^2*d)","A",4,3,21,0.1429,1,"{2721, 1647, 1629}"
183,1,109,0,0.0956814,"\int \frac{\tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]/(a + b*Sin[c + d*x])^2,x]","-\frac{a}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\left(a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}","-\frac{a}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\left(a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) + ((a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - a/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2721, 801}"
184,1,53,0,0.0526635,"\int \frac{\cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]/(a + b*Sin[c + d*x])^2,x]","-\frac{\log (a+b \sin (c+d x))}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sin (c+d x))}","-\frac{\log (a+b \sin (c+d x))}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sin (c+d x))}",1,"Log[Sin[c + d*x]]/(a^2*d) - Log[a + b*Sin[c + d*x]]/(a^2*d) + 1/(a*d*(a + b*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2721, 44}"
185,1,114,0,0.1147958,"\int \frac{\cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","-\frac{a^2-b^2}{a^3 d (a+b \sin (c+d x))}-\frac{\left(a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(a^2-3 b^2\right) \log (a+b \sin (c+d x))}{a^4 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{a^2-b^2}{a^3 d (a+b \sin (c+d x))}-\frac{\left(a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(a^2-3 b^2\right) \log (a+b \sin (c+d x))}{a^4 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"(2*b*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^2*d) - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^2 - 3*b^2)*Log[a + b*Sin[c + d*x]])/(a^4*d) - (a^2 - b^2)/(a^3*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2721, 894}"
186,1,188,0,0.1814682,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right)^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}","\frac{\left(a^2-b^2\right)^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}",1,"(-4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2721, 894}"
187,1,333,0,0.6266067,"\int \frac{\tan ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{8 a^3 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{a^4 b \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{(3 a+b) \cos (c+d x)}{4 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{12 d (a+b)^2 (1-\sin (c+d x))}+\frac{(3 a-b) \cos (c+d x)}{4 d (a-b)^3 (\sin (c+d x)+1)}-\frac{\cos (c+d x)}{12 d (a-b)^2 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^2 (1-\sin (c+d x))^2}-\frac{\cos (c+d x)}{12 d (a-b)^2 (\sin (c+d x)+1)^2}","\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{8 a^3 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{a^4 b \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{(3 a+b) \cos (c+d x)}{4 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{12 d (a+b)^2 (1-\sin (c+d x))}+\frac{(3 a-b) \cos (c+d x)}{4 d (a-b)^3 (\sin (c+d x)+1)}-\frac{\cos (c+d x)}{12 d (a-b)^2 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^2 (1-\sin (c+d x))^2}-\frac{\cos (c+d x)}{12 d (a-b)^2 (\sin (c+d x)+1)^2}",1,"(2*a^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (8*a^3*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(12*(a + b)^2*d*(1 - Sin[c + d*x])^2) + Cos[c + d*x]/(12*(a + b)^2*d*(1 - Sin[c + d*x])) - ((3*a + b)*Cos[c + d*x])/(4*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^2*d*(1 + Sin[c + d*x])^2) - Cos[c + d*x]/(12*(a - b)^2*d*(1 + Sin[c + d*x])) + ((3*a - b)*Cos[c + d*x])/(4*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^4*b*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",16,8,21,0.3810,1,"{2731, 2650, 2648, 2664, 12, 2660, 618, 204}"
188,1,200,0,0.3010283,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{4 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{4 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(-2*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (4*a*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",12,7,21,0.3333,1,"{2731, 2648, 2664, 12, 2660, 618, 204}"
189,1,115,0,0.4525335,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \sqrt{a^2-b^2}}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \sqrt{a^2-b^2}}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}",1,"(-2*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*Cot[c + d*x])/(a^2*d) + Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))","A",8,7,21,0.3333,1,"{2723, 3056, 3001, 3770, 2660, 618, 204}"
190,1,238,0,0.6952981,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{2 \left(-5 a^2 b^2+a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}","\frac{2 \left(-5 a^2 b^2+a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}",1,"(2*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) - (b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((7*a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) - ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^3*b*d) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x]))","A",8,7,21,0.3333,1,"{2724, 3055, 3001, 3770, 2660, 618, 204}"
191,1,424,0,1.4934229,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-6 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}-\frac{\left(-135 a^2 b^2+38 a^4+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{b \left(-40 a^2 b^2+15 a^4+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(-82 a^2 b^2+15 a^4+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{\left(-17 a^2 b^2+4 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(-12 a^2 b^2+2 a^4+9 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}","-\frac{2 \left(a^2-6 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}-\frac{\left(-135 a^2 b^2+38 a^4+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{b \left(-40 a^2 b^2+15 a^4+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(-82 a^2 b^2+15 a^4+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{\left(-17 a^2 b^2+4 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(-12 a^2 b^2+2 a^4+9 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}",1,"(-2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) + (b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*ArcTanh[Cos[c + d*x]])/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*Cot[c + d*x])/(15*a^6*d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^3*b^2*d*(a + b*Sin[c + d*x])) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x]))","A",10,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
192,1,321,0,0.8774219,"\int \frac{\tan ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","-\frac{a^5}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{a^4 \left(a^2+5 b^2\right)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a^3 \left(13 a^2 b^2+a^4+10 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{\left(8 a^2-5 a b-b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac{\left(8 a^2+5 a b-b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^4(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \sin (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) \left(8 a^3 \left(a^2+5 b^2\right)-b \left(22 a^2 b^2+27 a^4-b^4\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}","-\frac{a^5}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{a^4 \left(a^2+5 b^2\right)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a^3 \left(13 a^2 b^2+a^4+10 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{\left(8 a^2-5 a b-b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac{\left(8 a^2+5 a b-b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^4(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \sin (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) \left(8 a^3 \left(a^2+5 b^2\right)-b \left(22 a^2 b^2+27 a^4-b^4\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}",1,"-((8*a^2 - 5*a*b - b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^5*d) - ((8*a^2 + 5*a*b - b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^5*d) + (a^3*(a^4 + 13*a^2*b^2 + 10*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - a^5/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) - (a^4*(a^2 + 5*b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*(a^2 - b^2)^3*d) - (Sec[c + d*x]^2*(8*a^3*(a^2 + 5*b^2) - b*(27*a^4 + 22*a^2*b^2 - b^4)*Sin[c + d*x]))/(8*(a^2 - b^2)^4*d)","A",5,3,21,0.1429,1,"{2721, 1647, 1629}"
193,1,232,0,0.4823812,"\int \frac{\tan ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","\frac{a^3}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{a^2 \left(a^2+3 b^2\right)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a \left(8 a^2 b^2+a^4+3 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}+\frac{\sec ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(2 a-b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{4 d (a-b)^4}","\frac{a^3}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{a^2 \left(a^2+3 b^2\right)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a \left(8 a^2 b^2+a^4+3 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}+\frac{\sec ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(2 a-b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{4 d (a-b)^4}",1,"((2*a - b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^4*d) + ((2*a + b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^4*d) - (a*(a^4 + 8*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) + a^3/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (a^2*(a^2 + 3*b^2))/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)","A",4,3,21,0.1429,1,"{2721, 1647, 1629}"
194,1,149,0,0.1321632,"\int \frac{\tan (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]/(a + b*Sin[c + d*x])^3,x]","-\frac{a}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{a^2+b^2}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a \left(a^2+3 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^3}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^3}","-\frac{a}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{a^2+b^2}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a \left(a^2+3 b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^3}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^3}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)^3*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)^3*d) + (a*(a^2 + 3*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - a/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a^2 + b^2)/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2721, 801}"
195,1,75,0,0.060479,"\int \frac{\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]/(a + b*Sin[c + d*x])^3,x]","\frac{1}{a^2 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+b \sin (c+d x))^2}","\frac{1}{a^2 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+b \sin (c+d x))^2}",1,"Log[Sin[c + d*x]]/(a^3*d) - Log[a + b*Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + b*Sin[c + d*x])^2) + 1/(a^2*d*(a + b*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2721, 44}"
196,1,145,0,0.1332936,"\int \frac{\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","-\frac{a^2-3 b^2}{a^4 d (a+b \sin (c+d x))}-\frac{a^2-b^2}{2 a^3 d (a+b \sin (c+d x))^2}-\frac{\left(a^2-6 b^2\right) \log (\sin (c+d x))}{a^5 d}+\frac{\left(a^2-6 b^2\right) \log (a+b \sin (c+d x))}{a^5 d}+\frac{3 b \csc (c+d x)}{a^4 d}-\frac{\csc ^2(c+d x)}{2 a^3 d}","-\frac{a^2-3 b^2}{a^4 d (a+b \sin (c+d x))}-\frac{a^2-b^2}{2 a^3 d (a+b \sin (c+d x))^2}-\frac{\left(a^2-6 b^2\right) \log (\sin (c+d x))}{a^5 d}+\frac{\left(a^2-6 b^2\right) \log (a+b \sin (c+d x))}{a^5 d}+\frac{3 b \csc (c+d x)}{a^4 d}-\frac{\csc ^2(c+d x)}{2 a^3 d}",1,"(3*b*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^3*d) - ((a^2 - 6*b^2)*Log[Sin[c + d*x]])/(a^5*d) + ((a^2 - 6*b^2)*Log[a + b*Sin[c + d*x]])/(a^5*d) - (a^2 - b^2)/(2*a^3*d*(a + b*Sin[c + d*x])^2) - (a^2 - 3*b^2)/(a^4*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2721, 894}"
197,1,221,0,0.2110092,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","\frac{-6 a^2 b^2+a^4+5 b^4}{a^6 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac{\left(a^2-3 b^2\right) \csc ^2(c+d x)}{a^5 d}-\frac{2 b \left(3 a^2-5 b^2\right) \csc (c+d x)}{a^6 d}+\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \log (\sin (c+d x))}{a^7 d}-\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}","\frac{-6 a^2 b^2+a^4+5 b^4}{a^6 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac{\left(a^2-3 b^2\right) \csc ^2(c+d x)}{a^5 d}-\frac{2 b \left(3 a^2-5 b^2\right) \csc (c+d x)}{a^6 d}+\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \log (\sin (c+d x))}{a^7 d}-\frac{\left(-12 a^2 b^2+a^4+15 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}",1,"(-2*b*(3*a^2 - 5*b^2)*Csc[c + d*x])/(a^6*d) + ((a^2 - 3*b^2)*Csc[c + d*x]^2)/(a^5*d) + (b*Csc[c + d*x]^3)/(a^4*d) - Csc[c + d*x]^4/(4*a^3*d) + ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]])/(a^7*d) - ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) + (a^2 - b^2)^2/(2*a^5*d*(a + b*Sin[c + d*x])^2) + (a^4 - 6*a^2*b^2 + 5*b^4)/(a^6*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2721, 894}"
198,1,474,0,0.8717673,"\int \frac{\tan ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{a^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{8 a^4 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{12 a^2 b^2 \left(a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{3 a^5 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a^4 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{4 a^3 b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}-\frac{3 a \cos (c+d x)}{4 d (a+b)^4 (1-\sin (c+d x))}+\frac{3 a \cos (c+d x)}{4 d (a-b)^4 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{12 d (a-b)^3 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^3 (1-\sin (c+d x))^2}-\frac{\cos (c+d x)}{12 d (a-b)^3 (\sin (c+d x)+1)^2}","\frac{a^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{8 a^4 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{12 a^2 b^2 \left(a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{3 a^5 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a^4 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{4 a^3 b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}-\frac{3 a \cos (c+d x)}{4 d (a+b)^4 (1-\sin (c+d x))}+\frac{3 a \cos (c+d x)}{4 d (a-b)^4 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{12 d (a-b)^3 (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 d (a+b)^3 (1-\sin (c+d x))^2}-\frac{\cos (c+d x)}{12 d (a-b)^3 (\sin (c+d x)+1)^2}",1,"(8*a^4*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (12*a^2*b^2*(a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (a^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + Cos[c + d*x]/(12*(a + b)^3*d*(1 - Sin[c + d*x])^2) - (3*a*Cos[c + d*x])/(4*(a + b)^4*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(12*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^3*d*(1 + Sin[c + d*x])^2) + (3*a*Cos[c + d*x])/(4*(a - b)^4*d*(1 + Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^4*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (3*a^5*b*Cos[c + d*x])/(2*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (4*a^3*b^3*Cos[c + d*x])/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x]))","A",22,9,21,0.4286,1,"{2731, 2650, 2648, 2664, 2754, 12, 2660, 618, 204}"
199,1,350,0,0.5388858,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","-\frac{a^2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{4 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^3 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{2 a b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","-\frac{a^2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{4 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^3 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{2 a b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(-4*a^2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (a^2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (2*b^2*(3*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^3*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*a*b^3*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",18,8,21,0.3810,1,"{2731, 2648, 2664, 2754, 12, 2660, 618, 204}"
200,1,202,0,0.7884529,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"-(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)*d)) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x])/(2*a^3*(a^2 - b^2)*d) + Cot[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",9,8,21,0.3810,1,"{2723, 3056, 3055, 3001, 3770, 2660, 618, 204}"
201,1,289,0,1.0715273,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{\left(-19 a^2 b^2+2 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 d}-\frac{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left(3 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}","\frac{\left(-19 a^2 b^2+2 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 d}-\frac{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left(3 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}",1,"((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^3*b*d*(a + b*Sin[c + d*x]))","A",9,7,21,0.3333,1,"{2724, 3055, 3001, 3770, 2660, 618, 204}"
202,1,492,0,2.1268812,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+2 a^4+42 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}-\frac{\left(-645 a^2 b^2+91 a^4+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{b \left(-200 a^2 b^2+45 a^4+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(-187 a^2 b^2+15 a^4+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(-79 a^2 b^2+8 a^4+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left(-54 a^2 b^2+4 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-60 a^2 b^2+5 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}","-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+2 a^4+42 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}-\frac{\left(-645 a^2 b^2+91 a^4+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{b \left(-200 a^2 b^2+45 a^4+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(-187 a^2 b^2+15 a^4+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(-79 a^2 b^2+8 a^4+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left(-54 a^2 b^2+4 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-60 a^2 b^2+5 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}",1,"-((Sqrt[a^2 - b^2]*(2*a^4 - 29*a^2*b^2 + 42*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d)) + (b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c + d*x])/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^6*b*d) - ((15*a^4 - 187*a^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + ((5*a^4 - 60*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3)/(20*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^4 - 54*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^4*b^2*d*(a + b*Sin[c + d*x]))","A",11,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
203,1,271,0,0.3799268,"\int (a+b \sin (e+f x))^3 (g \tan (e+f x))^p \, dx","Int[(a + b*Sin[e + f*x])^3*(g*Tan[e + f*x])^p,x]","\frac{3 a^2 b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}+\frac{a^3 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{3 a b^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{b^3 \sin ^3(e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+4}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{f g (p+4)}","\frac{3 a^2 b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}+\frac{a^3 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{3 a b^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}+\frac{b^3 \sin ^3(e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+4}{2};\frac{p+6}{2};\sin ^2(e+f x)\right)}{f g (p+4)}",1,"(a^3*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (3*a^2*b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p)) + (b^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (4 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*(g*Tan[e + f*x])^(1 + p))/(f*g*(4 + p)) + (3*a*b^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))","A",10,6,23,0.2609,1,"{2722, 3476, 364, 2602, 2577, 2591}"
204,1,186,0,0.242273,"\int (a+b \sin (e+f x))^2 (g \tan (e+f x))^p \, dx","Int[(a + b*Sin[e + f*x])^2*(g*Tan[e + f*x])^p,x]","\frac{a^2 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}+\frac{b^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}","\frac{a^2 (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}+\frac{b^2 (g \tan (e+f x))^{p+3} \, _2F_1\left(2,\frac{p+3}{2};\frac{p+5}{2};-\tan ^2(e+f x)\right)}{f g^3 (p+3)}",1,"(a^2*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (2*a*b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p)) + (b^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))","A",8,6,23,0.2609,1,"{2722, 3476, 364, 2602, 2577, 2591}"
205,1,129,0,0.1478654,"\int (a+b \sin (e+f x)) (g \tan (e+f x))^p \, dx","Int[(a + b*Sin[e + f*x])*(g*Tan[e + f*x])^p,x]","\frac{a (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}","\frac{a (g \tan (e+f x))^{p+1} \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};-\tan ^2(e+f x)\right)}{f g (p+1)}+\frac{b \sin (e+f x) \cos ^2(e+f x)^{\frac{p+1}{2}} (g \tan (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+2}{2};\frac{p+4}{2};\sin ^2(e+f x)\right)}{f g (p+2)}",1,"(a*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))/(f*g*(2 + p))","A",6,5,21,0.2381,1,"{2722, 3476, 364, 2602, 2577}"
206,0,0,0,0.0480098,"\int \frac{(g \tan (e+f x))^p}{a+b \sin (e+f x)} \, dx","Int[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x]),x]","\int \frac{(g \tan (e+f x))^p}{a+b \sin (e+f x)} \, dx","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-p/2} (g \tan (e+f x))^p F_1\left(\frac{1-p}{2};-\frac{p}{2},1;\frac{3-p}{2};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{f (p-1) \left(b^2-a^2\right)}+\frac{a g \sin ^2(e+f x)^{\frac{1-p}{2}} (g \tan (e+f x))^{p-1} \left(1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)^{\frac{p-1}{2}} \, _2F_1\left(\frac{1-p}{2},\frac{1-p}{2};\frac{3-p}{2};\frac{\cos ^2(e+f x)-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}{1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}\right)}{f (p-1) \left(a^2-b^2\right)}",1,"Defer[Int][(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x]), x]","F",0,0,0,0,-1,"{}"
207,0,0,0,0.045093,"\int \frac{(g \tan (e+f x))^p}{(a+b \sin (e+f x))^2} \, dx","Int[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^2,x]","\int \frac{(g \tan (e+f x))^p}{(a+b \sin (e+f x))^2} \, dx","-\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-q/2} (g \tan (e+f x))^q F_1\left(\frac{1-q}{2};-\frac{q}{2},2;\frac{3-q}{2};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{f (q-1) \left(a^2-b^2\right)^2}+\frac{a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-q-1)} \left(1-\cos ^2(e+f x)\right)^{\frac{q-1}{2}} (g \tan (e+f x))^q \left(1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)^{\frac{3-q}{2}+\frac{q-1}{2}-2} \left(\left(2 \left(a^2-b^2\right)+b^2 (q+1) \cos ^2(e+f x)\right) \Phi \left(-\frac{a^2 \cot ^2(e+f x)}{a^2-b^2},1,\frac{1-q}{2}\right)-b^2 (q-1) \cos ^2(e+f x) \Phi \left(-\frac{a^2 \cot ^2(e+f x)}{a^2-b^2},1,\frac{3-q}{2}\right)\right)}{2 f \left(a^2-b^2\right)^2 \left(b^2-a^2\right)}-\frac{a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-q-1)} (g \tan (e+f x))^q \left(1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)^{\frac{q-1}{2}} \, _2F_1\left(\frac{1-q}{2},\frac{1-q}{2};\frac{3-q}{2};\frac{\cos ^2(e+f x)-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}{1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}\right)}{f (q-1) \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-q-1)} (g \tan (e+f x))^q \left(1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)^{\frac{q-1}{2}} \, _2F_1\left(\frac{1-q}{2},\frac{1-q}{2};\frac{3-q}{2};\frac{\cos ^2(e+f x)-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}{1-\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}}\right)}{f (q-1) \left(a^2-b^2\right)^2}",1,"Defer[Int][(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^2, x]","F",0,0,0,0,-1,"{}"
208,0,0,0,0.0403311,"\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx","Int[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p,x]","\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx","\text{Int}\left((g \tan (e+f x))^p (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x]","A",0,0,0,0,-1,"{}"