1,1,78,0,0.0659966,"\int \frac{\sin ^4(x)}{i+\tan (x)} \, dx","Int[Sin[x]^4/(I + Tan[x]),x]","-\frac{i x}{16}-\frac{i}{8 (-\tan (x)+i)}-\frac{3 i}{16 (\tan (x)+i)}-\frac{1}{32 (-\tan (x)+i)^2}-\frac{5}{32 (\tan (x)+i)^2}+\frac{i}{24 (\tan (x)+i)^3}","-\frac{i x}{16}-\frac{i}{8 (-\tan (x)+i)}-\frac{3 i}{16 (\tan (x)+i)}-\frac{1}{32 (-\tan (x)+i)^2}-\frac{5}{32 (\tan (x)+i)^2}+\frac{i}{24 (\tan (x)+i)^3}",1,"(-I/16)*x - 1/(32*(I - Tan[x])^2) - (I/8)/(I - Tan[x]) + (I/24)/(I + Tan[x])^3 - 5/(32*(I + Tan[x])^2) - ((3*I)/16)/(I + Tan[x])","A",5,4,13,0.3077,1,"{3516, 848, 88, 203}"
2,1,29,0,0.1368833,"\int \frac{\sin ^3(x)}{i+\tan (x)} \, dx","Int[Sin[x]^3/(I + Tan[x]),x]","\frac{\sin ^5(x)}{5}-\frac{1}{5} i \cos ^5(x)+\frac{1}{3} i \cos ^3(x)","\frac{\sin ^5(x)}{5}-\frac{1}{5} i \cos ^5(x)+\frac{1}{3} i \cos ^3(x)",1,"(I/3)*Cos[x]^3 - (I/5)*Cos[x]^5 + Sin[x]^5/5","A",9,7,13,0.5385,1,"{3518, 3108, 3107, 2565, 14, 2564, 30}"
3,1,50,0,0.0539709,"\int \frac{\sin ^2(x)}{i+\tan (x)} \, dx","Int[Sin[x]^2/(I + Tan[x]),x]","-\frac{i x}{8}-\frac{i}{8 (-\tan (x)+i)}-\frac{i}{4 (\tan (x)+i)}-\frac{1}{8 (\tan (x)+i)^2}","-\frac{i x}{8}-\frac{i}{8 (-\tan (x)+i)}-\frac{i}{4 (\tan (x)+i)}-\frac{1}{8 (\tan (x)+i)^2}",1,"(-I/8)*x - (I/8)/(I - Tan[x]) - 1/(8*(I + Tan[x])^2) - (I/4)/(I + Tan[x])","A",5,4,13,0.3077,1,"{3516, 848, 88, 203}"
4,1,19,0,0.0893613,"\int \frac{\sin (x)}{i+\tan (x)} \, dx","Int[Sin[x]/(I + Tan[x]),x]","\frac{\sin ^3(x)}{3}+\frac{1}{3} i \cos ^3(x)","\frac{\sin ^3(x)}{3}+\frac{1}{3} i \cos ^3(x)",1,"(I/3)*Cos[x]^3 + Sin[x]^3/3","A",8,6,11,0.5455,1,"{3518, 3108, 3107, 2565, 30, 2564}"
5,1,16,0,0.0864813,"\int \frac{\csc (x)}{i+\tan (x)} \, dx","Int[Csc[x]/(I + Tan[x]),x]","\sin (x)-i \cos (x)+i \tanh ^{-1}(\cos (x))","\sin (x)-i \cos (x)+i \tanh ^{-1}(\cos (x))",1,"I*ArcTanh[Cos[x]] - I*Cos[x] + Sin[x]","A",8,7,11,0.6364,1,"{3518, 3108, 3107, 2637, 2592, 321, 206}"
6,1,18,0,0.0333998,"\int \frac{\csc ^2(x)}{i+\tan (x)} \, dx","Int[Csc[x]^2/(I + Tan[x]),x]","i x+i \cot (x)+\log (\tan (x))+\log (\cos (x))","i x+i \cot (x)+\log (\tan (x))+\log (\cos (x))",1,"I*x + I*Cot[x] + Log[Cos[x]] + Log[Tan[x]]","A",3,2,13,0.1538,1,"{3516, 44}"
7,1,24,0,0.1322603,"\int \frac{\csc ^3(x)}{i+\tan (x)} \, dx","Int[Csc[x]^3/(I + Tan[x]),x]","-\csc (x)-\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)","-\csc (x)-\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)",1,"(-I/2)*ArcTanh[Cos[x]] - Csc[x] + (I/2)*Cot[x]*Csc[x]","A",8,7,13,0.5385,1,"{3518, 3108, 3107, 2606, 8, 2611, 3770}"
8,1,19,0,0.038832,"\int \frac{\csc ^4(x)}{i+\tan (x)} \, dx","Int[Csc[x]^4/(I + Tan[x]),x]","-\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x)","-\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x)",1,"-Cot[x]^2/2 + (I/3)*Cot[x]^3","A",4,3,13,0.2308,1,"{3516, 848, 43}"
9,1,40,0,0.1507257,"\int \frac{\csc ^5(x)}{i+\tan (x)} \, dx","Int[Csc[x]^5/(I + Tan[x]),x]","-\frac{\csc ^3(x)}{3}-\frac{1}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{1}{8} i \cot (x) \csc (x)","-\frac{\csc ^3(x)}{3}-\frac{1}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{1}{8} i \cot (x) \csc (x)",1,"(-I/8)*ArcTanh[Cos[x]] - (I/8)*Cot[x]*Csc[x] - Csc[x]^3/3 + (I/4)*Cot[x]*Csc[x]^3","A",9,8,13,0.6154,1,"{3518, 3108, 3107, 2606, 30, 2611, 3768, 3770}"
10,1,37,0,0.0469125,"\int \frac{\csc ^6(x)}{i+\tan (x)} \, dx","Int[Csc[x]^6/(I + Tan[x]),x]","\frac{1}{5} i \cot ^5(x)-\frac{\cot ^4(x)}{4}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}","\frac{1}{5} i \cot ^5(x)-\frac{\cot ^4(x)}{4}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}",1,"-Cot[x]^2/2 + (I/3)*Cot[x]^3 - Cot[x]^4/4 + (I/5)*Cot[x]^5","A",4,3,13,0.2308,1,"{3516, 848, 75}"
11,1,101,0,0.0764194,"\int \sin ^5(c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]^5*(a + b*Tan[c + d*x]),x]","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d + (2*a*Cos[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]^5)/(5*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) - (b*Sin[c + d*x]^5)/(5*d)","A",8,5,19,0.2632,1,"{3517, 2633, 2592, 302, 206}"
12,1,83,0,0.1680746,"\int \sin ^4(c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]^4*(a + b*Tan[c + d*x]),x]","-\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \tan (c+d x))}{4 d}-\frac{\sin (c+d x) \cos (c+d x) (3 a+4 b \tan (c+d x))}{8 d}+\frac{3 a x}{8}-\frac{b \log (\cos (c+d x))}{d}","-\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \tan (c+d x))}{4 d}-\frac{\sin (c+d x) \cos (c+d x) (3 a+4 b \tan (c+d x))}{8 d}+\frac{3 a x}{8}-\frac{b \log (\cos (c+d x))}{d}",1,"(3*a*x)/8 - (b*Log[Cos[c + d*x]])/d - (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Tan[c + d*x]))/(4*d) - (Cos[c + d*x]*Sin[c + d*x]*(3*a + 4*b*Tan[c + d*x]))/(8*d)","A",6,4,19,0.2105,1,"{819, 635, 203, 260}"
13,1,69,0,0.0679413,"\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x]),x]","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d)","A",8,5,19,0.2632,1,"{3517, 2633, 2592, 302, 206}"
14,1,49,0,0.0844788,"\int \sin ^2(c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x]),x]","-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d}","-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d}",1,"(a*x)/2 - (b*Log[Cos[c + d*x]])/d - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x]))/(2*d)","A",5,4,19,0.2105,1,"{819, 635, 203, 260}"
15,1,37,0,0.0357625,"\int \sin (c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]*(a + b*Tan[c + d*x]),x]","-\frac{a \cos (c+d x)}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d - (b*Sin[c + d*x])/d","A",6,5,17,0.2941,1,"{3517, 2638, 2592, 321, 206}"
16,1,26,0,0.029023,"\int \csc (c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) + (b*ArcTanh[Sin[c + d*x]])/d","A",4,2,17,0.1176,1,"{3517, 3770}"
17,1,25,0,0.0783231,"\int \csc ^2(c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x]),x]","\frac{b \log (\tan (c+d x))}{d}-\frac{a \cot (c+d x)}{d}","\frac{b \log (\tan (c+d x))}{d}-\frac{a \cot (c+d x)}{d}",1,"-((a*Cot[c + d*x])/d) + (b*Log[Tan[c + d*x]])/d","A",3,1,19,0.05263,1,"{43}"
18,1,60,0,0.0679689,"\int \csc ^3(c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x]),x]","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"-(a*ArcTanh[Cos[c + d*x]])/(2*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",7,6,19,0.3158,1,"{3517, 3768, 3770, 2621, 321, 207}"
19,1,57,0,0.0879247,"\int \csc ^4(c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d) + (b*Log[Tan[c + d*x]])/d","A",3,1,19,0.05263,1,"{766}"
20,1,98,0,0.0922901,"\int \csc ^5(c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x]),x]","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Csc[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",9,6,19,0.3158,1,"{3517, 3768, 3770, 2621, 302, 207}"
21,1,87,0,0.1075778,"\int \csc ^6(c+d x) (a+b \tan (c+d x)) \, dx","Int[Csc[c + d*x]^6*(a + b*Tan[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}",1,"-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/d - (2*a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d) + (b*Log[Tan[c + d*x]])/d","A",3,1,19,0.05263,1,"{766}"
22,1,113,0,0.185813,"\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","\frac{3}{8} x \left(a^2-5 b^2\right)+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{b^2 \tan (c+d x)}{d}","\frac{3}{8} x \left(a^2-5 b^2\right)+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{b^2 \tan (c+d x)}{d}",1,"(3*(a^2 - 5*b^2)*x)/8 - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (Cos[c + d*x]^2*(7*b - 5*a*Tan[c + d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(4*d)","A",8,6,21,0.2857,1,"{3516, 1645, 1810, 635, 203, 260}"
23,1,122,0,0.1306934,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \cos ^3(c+d x)}{3 d}+\frac{2 b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \cos ^3(c+d x)}{3 d}+\frac{2 b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(2*a*b*ArcTanh[Sin[c + d*x]])/d - (a^2*Cos[c + d*x])/d + (2*b^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) - (b^2*Cos[c + d*x]^3)/(3*d) + (b^2*Sec[c + d*x])/d - (2*a*b*Sin[c + d*x])/d - (2*a*b*Sin[c + d*x]^3)/(3*d)","A",11,7,21,0.3333,1,"{3517, 2633, 2592, 302, 206, 2590, 270}"
24,1,76,0,0.113079,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","\frac{1}{2} x \left(a^2-3 b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac{3 b^2 \tan (c+d x)}{2 d}","\frac{1}{2} x \left(a^2-3 b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac{3 b^2 \tan (c+d x)}{2 d}",1,"((a^2 - 3*b^2)*x)/2 - (2*a*b*Log[Cos[c + d*x]])/d + (3*b^2*Tan[c + d*x])/(2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(2*d)","A",6,6,21,0.2857,1,"{3516, 1645, 774, 635, 203, 260}"
25,1,68,0,0.0736639,"\int \sin (c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Sin[c + d*x]*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(2*a*b*ArcTanh[Sin[c + d*x]])/d - (a^2*Cos[c + d*x])/d + (b^2*Cos[c + d*x])/d + (b^2*Sec[c + d*x])/d - (2*a*b*Sin[c + d*x])/d","A",9,7,19,0.3684,1,"{3517, 2638, 2592, 321, 206, 2590, 14}"
26,1,43,0,0.0563928,"\int \csc (c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-((a^2*ArcTanh[Cos[c + d*x]])/d) + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (b^2*Sec[c + d*x])/d","A",6,4,19,0.2105,1,"{3517, 3770, 2606, 8}"
27,1,42,0,0.0474087,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-((a^2*Cot[c + d*x])/d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d","A",3,2,21,0.09524,1,"{3516, 43}"
28,1,95,0,0.1176061,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{d}",1,"-(a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (b^2*ArcTanh[Cos[c + d*x]])/d + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Csc[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (b^2*Sec[c + d*x])/d","A",10,7,21,0.3333,1,"{3517, 3768, 3770, 2621, 321, 207, 2622}"
29,1,79,0,0.0700101,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-(((a^2 + b^2)*Cot[c + d*x])/d) - (a*b*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d","A",3,2,21,0.09524,1,"{3516, 894}"
30,1,165,0,0.1600818,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]","-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}","-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}",1,"(-3*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (3*b^2*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Csc[c + d*x])/d - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*b^2*Sec[c + d*x])/(2*d) - (b^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d)","A",13,9,21,0.4286,1,"{3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321}"
31,1,122,0,0.1012805,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]","-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{2 a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{2 a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-(((a^2 + 2*b^2)*Cot[c + d*x])/d) - (2*a*b*Cot[c + d*x]^2)/d - ((2*a^2 + b^2)*Cot[c + d*x]^3)/(3*d) - (a*b*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d","A",3,2,21,0.09524,1,"{3516, 948}"
32,1,205,0,0.1888733,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cos (c+d x)}{d}-\frac{a b^2 \cos ^3(c+d x)}{d}+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}","-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cos (c+d x)}{d}-\frac{a b^2 \cos ^3(c+d x)}{d}+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cos[c + d*x])/d + (6*a*b^2*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (a*b^2*Cos[c + d*x]^3)/d + (3*a*b^2*Sec[c + d*x])/d - (3*a^2*b*Sin[c + d*x])/d + (5*b^3*Sin[c + d*x])/(2*d) - (a^2*b*Sin[c + d*x]^3)/d + (5*b^3*Sin[c + d*x]^3)/(6*d) + (b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(2*d)","A",16,8,21,0.3810,1,"{3517, 2633, 2592, 302, 206, 2590, 270, 288}"
33,1,103,0,0.1445928,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} a x \left(a^2-9 b^2\right)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^3}{2 d}+\frac{b^3 \tan ^2(c+d x)}{d}","-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} a x \left(a^2-9 b^2\right)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^3}{2 d}+\frac{b^3 \tan ^2(c+d x)}{d}",1,"(a*(a^2 - 9*b^2)*x)/2 - (b*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (9*a*b^2*Tan[c + d*x])/(2*d) + (b^3*Tan[c + d*x]^2)/d - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(2*d)","A",7,6,21,0.2857,1,"{3516, 1645, 801, 635, 203, 260}"
34,1,133,0,0.1164818,"\int \sin (c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Sin[c + d*x]*(a + b*Tan[c + d*x])^3,x]","-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{3 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}","-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{3 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cos[c + d*x])/d + (3*a*b^2*Cos[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d - (3*a^2*b*Sin[c + d*x])/d + (3*b^3*Sin[c + d*x])/(2*d) + (b^3*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)","A",13,8,19,0.4211,1,"{3517, 2638, 2592, 321, 206, 2590, 14, 288}"
35,1,86,0,0.0856801,"\int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]*(a + b*Tan[c + d*x])^3,x]","\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-((a^3*ArcTanh[Cos[c + d*x]])/d) + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (b^3*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",8,5,19,0.2632,1,"{3517, 3770, 2606, 8, 2611}"
36,1,64,0,0.0525311,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","\frac{3 a^2 b \log (\tan (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}","\frac{3 a^2 b \log (\tan (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"-((a^3*Cot[c + d*x])/d) + (3*a^2*b*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{3516, 43}"
37,1,141,0,0.1349043,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}","-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-(a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*b^2*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Csc[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",12,7,21,0.3333,1,"{3517, 3768, 3770, 2621, 321, 207, 2622}"
38,1,113,0,0.0860254,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]","-\frac{a \left(a^2+3 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}","-\frac{a \left(a^2+3 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"-((a*(a^2 + 3*b^2)*Cot[c + d*x])/d) - (3*a^2*b*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d) + (b*(3*a^2 + b^2)*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{3516, 894}"
39,1,229,0,0.2080383,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]","-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}","-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}",1,"(-3*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (9*a*b^2*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Csc[c + d*x])/d - (3*b^3*Csc[c + d*x])/(2*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*b*Csc[c + d*x]^3)/d - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (9*a*b^2*Sec[c + d*x])/(2*d) - (3*a*b^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (b^3*Csc[c + d*x]*Sec[c + d*x]^2)/(2*d)","A",17,9,21,0.4286,1,"{3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321}"
40,1,167,0,0.134132,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]","-\frac{a \left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{b \left(6 a^2+b^2\right) \cot ^2(c+d x)}{2 d}-\frac{a \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^4(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}","-\frac{a \left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{b \left(6 a^2+b^2\right) \cot ^2(c+d x)}{2 d}-\frac{a \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^4(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"-((a*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (b*(6*a^2 + b^2)*Cot[c + d*x]^2)/(2*d) - (a*(2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - (3*a^2*b*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{3516, 948}"
41,1,275,0,0.2476161,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cos (c+d x)}{d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{3 b^4 \sec (c+d x)}{d}","-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cos (c+d x)}{d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{3 b^4 \sec (c+d x)}{d}",1,"(4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (12*a^2*b^2*Cos[c + d*x])/d - (3*b^4*Cos[c + d*x])/d + (a^4*Cos[c + d*x]^3)/(3*d) - (2*a^2*b^2*Cos[c + d*x]^3)/d + (b^4*Cos[c + d*x]^3)/(3*d) + (6*a^2*b^2*Sec[c + d*x])/d - (3*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*b*Sin[c + d*x])/d + (10*a*b^3*Sin[c + d*x])/d - (4*a^3*b*Sin[c + d*x]^3)/(3*d) + (10*a*b^3*Sin[c + d*x]^3)/(3*d) + (2*a*b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/d","A",19,8,21,0.3810,1,"{3517, 2633, 2592, 302, 206, 2590, 270, 288}"
42,1,139,0,0.1749745,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","\frac{b^2 \left(18 a^2-5 b^2\right) \tan (c+d x)}{2 d}-\frac{4 a b \left(a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} x \left(-18 a^2 b^2+a^4+5 b^4\right)+\frac{4 a b^3 \tan ^2(c+d x)}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}","\frac{b^2 \left(18 a^2-5 b^2\right) \tan (c+d x)}{2 d}-\frac{4 a b \left(a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} x \left(-18 a^2 b^2+a^4+5 b^4\right)+\frac{4 a b^3 \tan ^2(c+d x)}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}",1,"((a^4 - 18*a^2*b^2 + 5*b^4)*x)/2 - (4*a*b*(a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (b^2*(18*a^2 - 5*b^2)*Tan[c + d*x])/(2*d) + (4*a*b^3*Tan[c + d*x]^2)/d + (5*b^4*Tan[c + d*x]^3)/(6*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^4)/(2*d)","A",7,6,21,0.2857,1,"{3516, 1645, 801, 635, 203, 260}"
43,1,180,0,0.15661,"\int \sin (c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Sin[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{6 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \cos (c+d x)}{d}+\frac{6 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}-\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{2 b^4 \sec (c+d x)}{d}","\frac{6 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \cos (c+d x)}{d}+\frac{6 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}-\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{2 b^4 \sec (c+d x)}{d}",1,"(4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (6*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (6*a^2*b^2*Cos[c + d*x])/d - (b^4*Cos[c + d*x])/d + (6*a^2*b^2*Sec[c + d*x])/d - (2*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*b*Sin[c + d*x])/d + (6*a*b^3*Sin[c + d*x])/d + (2*a*b^3*Sin[c + d*x]*Tan[c + d*x]^2)/d","A",16,9,19,0.4737,1,"{3517, 2638, 2592, 321, 206, 2590, 14, 288, 270}"
44,1,118,0,0.1142193,"\int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{b^4 \sec (c+d x)}{d}","\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{b^4 \sec (c+d x)}{d}",1,"-((a^4*ArcTanh[Cos[c + d*x]])/d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b^3*ArcTanh[Sin[c + d*x]])/d + (6*a^2*b^2*Sec[c + d*x])/d - (b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) + (2*a*b^3*Sec[c + d*x]*Tan[c + d*x])/d","A",10,5,19,0.2632,1,"{3517, 3770, 2606, 8, 2611}"
45,1,83,0,0.0581838,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","\frac{6 a^2 b^2 \tan (c+d x)}{d}+\frac{4 a^3 b \log (\tan (c+d x))}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}","\frac{6 a^2 b^2 \tan (c+d x)}{d}+\frac{4 a^3 b \log (\tan (c+d x))}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-((a^4*Cot[c + d*x])/d) + (4*a^3*b*Log[Tan[c + d*x]])/d + (6*a^2*b^2*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{3516, 43}"
46,1,161,0,0.1574332,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}","\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}",1,"-(a^4*ArcTanh[Cos[c + d*x]])/(2*d) - (6*a^2*b^2*ArcTanh[Cos[c + d*x]])/d + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (2*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (a^4*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (6*a^2*b^2*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) + (2*a*b^3*Sec[c + d*x]*Tan[c + d*x])/d","A",14,9,21,0.4286,1,"{3517, 3768, 3770, 2621, 321, 207, 2622, 2606, 30}"
47,1,137,0,0.1042266,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^4,x]","\frac{b^2 \left(6 a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{4 a b \left(a^2+b^2\right) \log (\tan (c+d x))}{d}-\frac{2 a^3 b \cot ^2(c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}","\frac{b^2 \left(6 a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{4 a b \left(a^2+b^2\right) \log (\tan (c+d x))}{d}-\frac{2 a^3 b \cot ^2(c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-((a^2*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (2*a^3*b*Cot[c + d*x]^2)/d - (a^4*Cot[c + d*x]^3)/(3*d) + (4*a*b*(a^2 + b^2)*Log[Tan[c + d*x]])/d + (b^2*(6*a^2 + b^2)*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{3516, 894}"
48,1,274,0,0.2408827,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]","\frac{9 a^2 b^2 \sec (c+d x)}{d}-\frac{9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{6 a b^3 \csc (c+d x)}{d}+\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}+\frac{b^4 \sec (c+d x)}{d}-\frac{b^4 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{9 a^2 b^2 \sec (c+d x)}{d}-\frac{9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{6 a b^3 \csc (c+d x)}{d}+\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}+\frac{b^4 \sec (c+d x)}{d}-\frac{b^4 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-3*a^4*ArcTanh[Cos[c + d*x]])/(8*d) - (9*a^2*b^2*ArcTanh[Cos[c + d*x]])/d - (b^4*ArcTanh[Cos[c + d*x]])/d + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (6*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (6*a*b^3*Csc[c + d*x])/d - (3*a^4*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (9*a^2*b^2*Sec[c + d*x])/d + (b^4*Sec[c + d*x])/d - (3*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*x])/d + (2*a*b^3*Csc[c + d*x]*Sec[c + d*x]^2)/d + (b^4*Sec[c + d*x]^3)/(3*d)","A",21,9,21,0.4286,1,"{3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321}"
49,1,194,0,0.1570773,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^4,x]","\frac{2 b^2 \left(3 a^2+b^2\right) \tan (c+d x)}{d}-\frac{2 a^2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{2 a b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{d}-\frac{\left(12 a^2 b^2+a^4+b^4\right) \cot (c+d x)}{d}+\frac{4 a b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{a^3 b \cot ^4(c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}","\frac{2 b^2 \left(3 a^2+b^2\right) \tan (c+d x)}{d}-\frac{2 a^2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{2 a b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{d}-\frac{\left(12 a^2 b^2+a^4+b^4\right) \cot (c+d x)}{d}+\frac{4 a b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{a^3 b \cot ^4(c+d x)}{d}-\frac{a^4 \cot ^5(c+d x)}{5 d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-(((a^4 + 12*a^2*b^2 + b^4)*Cot[c + d*x])/d) - (2*a*b*(2*a^2 + b^2)*Cot[c + d*x]^2)/d - (2*a^2*(a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - (a^3*b*Cot[c + d*x]^4)/d - (a^4*Cot[c + d*x]^5)/(5*d) + (4*a*b*(a^2 + 2*b^2)*Log[Tan[c + d*x]])/d + (2*b^2*(3*a^2 + b^2)*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{3516, 948}"
50,1,402,0,0.3055422,"\int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx","Int[Csc[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]","\frac{45 a^2 b^2 \sec (c+d x)}{4 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc (c+d x)}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}","\frac{45 a^2 b^2 \sec (c+d x)}{4 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc (c+d x)}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}",1,"(-5*a^4*ArcTanh[Cos[c + d*x]])/(16*d) - (45*a^2*b^2*ArcTanh[Cos[c + d*x]])/(4*d) - (5*b^4*ArcTanh[Cos[c + d*x]])/(2*d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (10*a*b^3*Csc[c + d*x])/d - (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (10*a*b^3*Csc[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (4*a^3*b*Csc[c + d*x]^5)/(5*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (45*a^2*b^2*Sec[c + d*x])/(4*d) + (5*b^4*Sec[c + d*x])/(2*d) - (15*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*x])/(4*d) - (3*a^2*b^2*Csc[c + d*x]^4*Sec[c + d*x])/(2*d) + (2*a*b^3*Csc[c + d*x]^3*Sec[c + d*x]^2)/d + (5*b^4*Sec[c + d*x]^3)/(6*d) - (b^4*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d)","A",25,9,21,0.4286,1,"{3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321}"
51,1,274,0,0.3539015,"\int \frac{\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]^5/(a + b*Tan[c + d*x]),x]","\frac{a^2 b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}+\frac{b \sin ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{a^4 b \sin (c+d x)}{d \left(a^2+b^2\right)^3}-\frac{a \cos ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{2 a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a b^2 \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}+\frac{a^3 b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^3}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^5 b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{7/2}}","\frac{a^2 b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}+\frac{b \sin ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{a^4 b \sin (c+d x)}{d \left(a^2+b^2\right)^3}-\frac{a \cos ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{2 a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a b^2 \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}+\frac{a^3 b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^3}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^5 b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{7/2}}",1,"(a^5*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d) + (a^3*b^2*Cos[c + d*x])/((a^2 + b^2)^3*d) + (a*b^2*Cos[c + d*x])/((a^2 + b^2)^2*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) - (a*b^2*Cos[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (2*a*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) - (a*Cos[c + d*x]^5)/(5*(a^2 + b^2)*d) + (a^4*b*Sin[c + d*x])/((a^2 + b^2)^3*d) + (a^2*b*Sin[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (b*Sin[c + d*x]^5)/(5*(a^2 + b^2)*d)","A",13,9,21,0.4286,1,"{3518, 3109, 2564, 30, 2633, 3099, 3074, 206, 2638}"
52,1,158,0,0.3389488,"\int \frac{\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x]),x]","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2\right) \tan (c+d x)+4 b \left(2 a^2+b^2\right)\right)}{8 d \left(a^2+b^2\right)^2}+\frac{a^4 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(-6 a^2 b^2+3 a^4-b^4\right)}{8 \left(a^2+b^2\right)^3}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2\right) \tan (c+d x)+4 b \left(2 a^2+b^2\right)\right)}{8 d \left(a^2+b^2\right)^2}+\frac{a^4 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(-6 a^2 b^2+3 a^4-b^4\right)}{8 \left(a^2+b^2\right)^3}",1,"(a*(3*a^4 - 6*a^2*b^2 - b^4)*x)/(8*(a^2 + b^2)^3) + (a^4*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d) - (Cos[c + d*x]^2*(4*b*(2*a^2 + b^2) + a*(5*a^2 + b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)","A",8,6,21,0.2857,1,"{3516, 1647, 801, 635, 203, 260}"
53,1,168,0,0.2229897,"\int \frac{\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]^3/(a + b*Tan[c + d*x]),x]","\frac{b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a^2 b \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}","\frac{b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a^2 b \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}",1,"(a^3*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d) + (a*b^2*Cos[c + d*x])/((a^2 + b^2)^2*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) + (a*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a^2*b*Sin[c + d*x])/((a^2 + b^2)^2*d) + (b*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d)","A",10,9,21,0.4286,1,"{3518, 3109, 2564, 30, 2633, 3099, 3074, 206, 2638}"
54,1,94,0,0.1632177,"\int \frac{\sin ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]^2/(a + b*Tan[c + d*x]),x]","-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right)}+\frac{a^2 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{a x \left(a^2-b^2\right)}{2 \left(a^2+b^2\right)^2}","-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right)}+\frac{a^2 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{a x \left(a^2-b^2\right)}{2 \left(a^2+b^2\right)^2}",1,"(a*(a^2 - b^2)*x)/(2*(a^2 + b^2)^2) + (a^2*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(2*(a^2 + b^2)*d)","A",7,6,21,0.2857,1,"{3516, 1647, 801, 635, 203, 260}"
55,1,90,0,0.1057159,"\int \frac{\sin (c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{b \sin (c+d x)}{d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}","\frac{b \sin (c+d x)}{d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}",1,"(a*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) + (b*Sin[c + d*x])/((a^2 + b^2)*d)","A",6,6,19,0.3158,1,"{3518, 3109, 2637, 2638, 3074, 206}"
56,1,66,0,0.1287643,"\int \frac{\csc (c+d x)}{a+b \tan (c+d x)} \, dx","Int[Csc[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{a d \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{a d \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(ArcTanh[Cos[c + d*x]]/(a*d)) + (b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)","A",6,5,19,0.2632,1,"{3518, 3110, 3770, 3074, 206}"
57,1,50,0,0.0602185,"\int \frac{\csc ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Csc[c + d*x]^2/(a + b*Tan[c + d*x]),x]","-\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}","-\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"-(Cot[c + d*x]/(a*d)) - (b*Log[Tan[c + d*x]])/(a^2*d) + (b*Log[a + b*Tan[c + d*x]])/(a^2*d)","A",3,2,21,0.09524,1,"{3516, 44}"
58,1,122,0,0.3072341,"\int \frac{\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Csc[c + d*x]^3/(a + b*Tan[c + d*x]),x]","-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{a^3 d}+\frac{b \csc (c+d x)}{a^2 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) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^3*d) + (b*Csc[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",15,11,21,0.5238,1,"{3518, 3110, 3768, 3770, 2621, 321, 207, 2622, 3104, 3074, 206}"
59,1,108,0,0.1044534,"\int \frac{\csc ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Csc[c + d*x]^4/(a + b*Tan[c + d*x]),x]","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{a^3 d}-\frac{b \left(a^2+b^2\right) \log (\tan (c+d x))}{a^4 d}+\frac{b \left(a^2+b^2\right) \log (a+b \tan (c+d x))}{a^4 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{a^3 d}-\frac{b \left(a^2+b^2\right) \log (\tan (c+d x))}{a^4 d}+\frac{b \left(a^2+b^2\right) \log (a+b \tan (c+d x))}{a^4 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}",1,"-(((a^2 + b^2)*Cot[c + d*x])/(a^3*d)) + (b*Cot[c + d*x]^2)/(2*a^2*d) - Cot[c + d*x]^3/(3*a*d) - (b*(a^2 + b^2)*Log[Tan[c + d*x]])/(a^4*d) + (b*(a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a^4*d)","A",3,2,21,0.09524,1,"{3516, 894}"
60,1,169,0,0.1507826,"\int \frac{\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x]),x]","-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 a^3 d}+\frac{b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2+b^2\right)^2 \cot (c+d x)}{a^5 d}-\frac{b \left(a^2+b^2\right)^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))}{a^6 d}+\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}","-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 a^3 d}+\frac{b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2+b^2\right)^2 \cot (c+d x)}{a^5 d}-\frac{b \left(a^2+b^2\right)^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))}{a^6 d}+\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"-(((a^2 + b^2)^2*Cot[c + d*x])/(a^5*d)) + (b*(2*a^2 + b^2)*Cot[c + d*x]^2)/(2*a^4*d) - ((2*a^2 + b^2)*Cot[c + d*x]^3)/(3*a^3*d) + (b*Cot[c + d*x]^4)/(4*a^2*d) - Cot[c + d*x]^5/(5*a*d) - (b*(a^2 + b^2)^2*Log[Tan[c + d*x]])/(a^6*d) + (b*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])/(a^6*d)","A",3,2,21,0.09524,1,"{3516, 894}"
61,1,297,0,0.9136414,"\int \frac{\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Sin[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","-\frac{a^6 b}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{\cos ^6(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{6 d \left(a^2+b^2\right)^2}+\frac{\cos ^4(c+d x) \left(\left(-18 a^2 b^2+13 a^4-7 b^4\right) \tan (c+d x)+12 a b \left(3 a^2+b^2\right)\right)}{24 d \left(a^2+b^2\right)^3}-\frac{\cos ^2(c+d x) \left(\left(-43 a^4 b^2-7 a^2 b^4+11 a^6-b^6\right) \tan (c+d x)+48 a^5 b\right)}{16 d \left(a^2+b^2\right)^4}+\frac{2 a^5 b \left(a^2-3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{x \left(-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+5 a^8+b^8\right)}{16 \left(a^2+b^2\right)^5}","-\frac{a^6 b}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{\cos ^6(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{6 d \left(a^2+b^2\right)^2}+\frac{\cos ^4(c+d x) \left(\left(-18 a^2 b^2+13 a^4-7 b^4\right) \tan (c+d x)+12 a b \left(3 a^2+b^2\right)\right)}{24 d \left(a^2+b^2\right)^3}-\frac{\cos ^2(c+d x) \left(\left(-43 a^4 b^2-7 a^2 b^4+11 a^6-b^6\right) \tan (c+d x)+48 a^5 b\right)}{16 d \left(a^2+b^2\right)^4}+\frac{2 a^5 b \left(a^2-3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{x \left(-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+5 a^8+b^8\right)}{16 \left(a^2+b^2\right)^5}",1,"((5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*x)/(16*(a^2 + b^2)^5) + (2*a^5*b*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^6*b)/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^6*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^2*d) + (Cos[c + d*x]^4*(12*a*b*(3*a^2 + b^2) + (13*a^4 - 18*a^2*b^2 - 7*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^3*d) - (Cos[c + d*x]^2*(48*a^5*b + (11*a^6 - 43*a^4*b^2 - 7*a^2*b^4 - b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^4*d)","A",9,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
62,1,217,0,0.5619448,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","-\frac{a^4 b}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{4 d \left(a^2+b^2\right)^2}-\frac{\cos ^2(c+d x) \left(\left(-12 a^2 b^2+5 a^4-b^4\right) \tan (c+d x)+16 a^3 b\right)}{8 d \left(a^2+b^2\right)^3}+\frac{2 a^3 b \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right)}{8 \left(a^2+b^2\right)^4}","-\frac{a^4 b}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{4 d \left(a^2+b^2\right)^2}-\frac{\cos ^2(c+d x) \left(\left(-12 a^2 b^2+5 a^4-b^4\right) \tan (c+d x)+16 a^3 b\right)}{8 d \left(a^2+b^2\right)^3}+\frac{2 a^3 b \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right)}{8 \left(a^2+b^2\right)^4}",1,"((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*x)/(8*(a^2 + b^2)^4) + (2*a^3*b*(a^2 - 2*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^4*b)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(16*a^3*b + (5*a^4 - 12*a^2*b^2 - b^4)*Tan[c + d*x]))/(8*(a^2 + b^2)^3*d)","A",8,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
63,1,148,0,0.2949043,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{2 d \left(a^2+b^2\right)^2}+\frac{2 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(-6 a^2 b^2+a^4+b^4\right)}{2 \left(a^2+b^2\right)^3}","-\frac{a^2 b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{2 d \left(a^2+b^2\right)^2}+\frac{2 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(-6 a^2 b^2+a^4+b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"((a^4 - 6*a^2*b^2 + b^4)*x)/(2*(a^2 + b^2)^3) + (2*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (a^2*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(2*(a^2 + b^2)^2*d)","A",7,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
64,1,72,0,0.0654904,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","-\frac{b}{a^2 d (a+b \tan (c+d x))}-\frac{2 b \log (\tan (c+d x))}{a^3 d}+\frac{2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}","-\frac{b}{a^2 d (a+b \tan (c+d x))}-\frac{2 b \log (\tan (c+d x))}{a^3 d}+\frac{2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}",1,"-(Cot[c + d*x]/(a^2*d)) - (2*b*Log[Tan[c + d*x]])/(a^3*d) + (2*b*Log[a + b*Tan[c + d*x]])/(a^3*d) - b/(a^2*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 44}"
65,1,140,0,0.1224969,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","-\frac{b \left(a^2+b^2\right)}{a^4 d (a+b \tan (c+d x))}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{a^4 d}-\frac{2 b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{a^5 d}+\frac{2 b \left(a^2+2 b^2\right) \log (a+b \tan (c+d x))}{a^5 d}+\frac{b \cot ^2(c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}","-\frac{b \left(a^2+b^2\right)}{a^4 d (a+b \tan (c+d x))}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{a^4 d}-\frac{2 b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{a^5 d}+\frac{2 b \left(a^2+2 b^2\right) \log (a+b \tan (c+d x))}{a^5 d}+\frac{b \cot ^2(c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}",1,"-(((a^2 + 3*b^2)*Cot[c + d*x])/(a^4*d)) + (b*Cot[c + d*x]^2)/(a^3*d) - Cot[c + d*x]^3/(3*a^2*d) - (2*b*(a^2 + 2*b^2)*Log[Tan[c + d*x]])/(a^5*d) + (2*b*(a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]])/(a^5*d) - (b*(a^2 + b^2))/(a^4*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
66,1,219,0,0.1983339,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","-\frac{b \left(a^2+b^2\right)^2}{a^6 d (a+b \tan (c+d x))}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^4 d}+\frac{2 b \left(a^2+b^2\right) \cot ^2(c+d x)}{a^5 d}-\frac{\left(a^2+b^2\right) \left(a^2+5 b^2\right) \cot (c+d x)}{a^6 d}-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}+\frac{b \cot ^4(c+d x)}{2 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}","-\frac{b \left(a^2+b^2\right)^2}{a^6 d (a+b \tan (c+d x))}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^4 d}+\frac{2 b \left(a^2+b^2\right) \cot ^2(c+d x)}{a^5 d}-\frac{\left(a^2+b^2\right) \left(a^2+5 b^2\right) \cot (c+d x)}{a^6 d}-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}+\frac{b \cot ^4(c+d x)}{2 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}",1,"-(((a^2 + b^2)*(a^2 + 5*b^2)*Cot[c + d*x])/(a^6*d)) + (2*b*(a^2 + b^2)*Cot[c + d*x]^2)/(a^5*d) - ((2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*a^4*d) + (b*Cot[c + d*x]^4)/(2*a^3*d) - Cot[c + d*x]^5/(5*a^2*d) - (2*b*(a^2 + b^2)*(a^2 + 3*b^2)*Log[Tan[c + d*x]])/(a^7*d) + (2*b*(a^2 + b^2)*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^7*d) - (b*(a^2 + b^2)^2)/(a^6*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
67,1,382,0,1.4318173,"\int \frac{\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Sin[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","-\frac{a^6 b}{2 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{2 a^5 b \left(a^2-3 b^2\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left(\left(-119 a^4 b^2+65 a^2 b^4+11 a^6+3 b^6\right) \tan (c+d x)+24 a^3 b \left(3 a^2-5 b^2\right)\right)}{16 d \left(a^2+b^2\right)^5}-\frac{\cos ^6(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{6 d \left(a^2+b^2\right)^3}+\frac{\cos ^4(c+d x) \left(a \left(-62 a^2 b^2+13 a^4-3 b^4\right) \tan (c+d x)+6 b \left(-4 a^2 b^2+9 a^4-b^4\right)\right)}{24 d \left(a^2+b^2\right)^4}+\frac{a^4 b \left(-22 a^2 b^2+3 a^4+15 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}+\frac{a x \left(-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6+5 a^8-3 b^8\right)}{16 \left(a^2+b^2\right)^6}","-\frac{a^6 b}{2 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{2 a^5 b \left(a^2-3 b^2\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left(\left(-119 a^4 b^2+65 a^2 b^4+11 a^6+3 b^6\right) \tan (c+d x)+24 a^3 b \left(3 a^2-5 b^2\right)\right)}{16 d \left(a^2+b^2\right)^5}-\frac{\cos ^6(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{6 d \left(a^2+b^2\right)^3}+\frac{\cos ^4(c+d x) \left(a \left(-62 a^2 b^2+13 a^4-3 b^4\right) \tan (c+d x)+6 b \left(-4 a^2 b^2+9 a^4-b^4\right)\right)}{24 d \left(a^2+b^2\right)^4}+\frac{a^4 b \left(-22 a^2 b^2+3 a^4+15 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}+\frac{a x \left(-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6+5 a^8-3 b^8\right)}{16 \left(a^2+b^2\right)^6}",1,"(a*(5*a^8 - 180*a^6*b^2 + 390*a^4*b^4 - 68*a^2*b^6 - 3*b^8)*x)/(16*(a^2 + b^2)^6) + (a^4*b*(3*a^4 - 22*a^2*b^2 + 15*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^6*b)/(2*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^2) - (2*a^5*b*(a^2 - 3*b^2))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^6*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(6*b*(9*a^4 - 4*a^2*b^2 - b^4) + a*(13*a^4 - 62*a^2*b^2 - 3*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^4*d) - (a*Cos[c + d*x]^2*(24*a^3*b*(3*a^2 - 5*b^2) + (11*a^6 - 119*a^4*b^2 + 65*a^2*b^4 + 3*b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^5*d)","A",9,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
68,1,285,0,0.8497479,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","-\frac{a^4 b}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left(\left(-34 a^2 b^2+5 a^4+9 b^4\right) \tan (c+d x)+24 a b \left(a^2-b^2\right)\right)}{8 d \left(a^2+b^2\right)^4}+\frac{\cos ^4(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{4 d \left(a^2+b^2\right)^3}+\frac{3 a^2 b \left(-5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{3 a x \left(-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right)}{8 \left(a^2+b^2\right)^5}","-\frac{a^4 b}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left(\left(-34 a^2 b^2+5 a^4+9 b^4\right) \tan (c+d x)+24 a b \left(a^2-b^2\right)\right)}{8 d \left(a^2+b^2\right)^4}+\frac{\cos ^4(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{4 d \left(a^2+b^2\right)^3}+\frac{3 a^2 b \left(-5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{3 a x \left(-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right)}{8 \left(a^2+b^2\right)^5}",1,"(3*a*(a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(8*(a^2 + b^2)^5) + (3*a^2*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^4*b)/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (2*a^3*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^3*d) - (a*Cos[c + d*x]^2*(24*a*b*(a^2 - b^2) + (5*a^4 - 34*a^2*b^2 + 9*b^4)*Tan[c + d*x]))/(8*(a^2 + b^2)^4*d)","A",8,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
69,1,206,0,0.3962461,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","-\frac{a^2 b}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{2 a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{2 d \left(a^2+b^2\right)^3}+\frac{b \left(-8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{a x \left(-14 a^2 b^2+a^4+9 b^4\right)}{2 \left(a^2+b^2\right)^4}","-\frac{a^2 b}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{2 a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{2 d \left(a^2+b^2\right)^3}+\frac{b \left(-8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{a x \left(-14 a^2 b^2+a^4+9 b^4\right)}{2 \left(a^2+b^2\right)^4}",1,"(a*(a^4 - 14*a^2*b^2 + 9*b^4)*x)/(2*(a^2 + b^2)^4) + (b*(3*a^4 - 8*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^2*b)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a*b*(a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(2*(a^2 + b^2)^3*d)","A",7,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
70,1,95,0,0.0774468,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","-\frac{2 b}{a^3 d (a+b \tan (c+d x))}-\frac{b}{2 a^2 d (a+b \tan (c+d x))^2}-\frac{3 b \log (\tan (c+d x))}{a^4 d}+\frac{3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}","-\frac{2 b}{a^3 d (a+b \tan (c+d x))}-\frac{b}{2 a^2 d (a+b \tan (c+d x))^2}-\frac{3 b \log (\tan (c+d x))}{a^4 d}+\frac{3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}",1,"-(Cot[c + d*x]/(a^3*d)) - (3*b*Log[Tan[c + d*x]])/(a^4*d) + (3*b*Log[a + b*Tan[c + d*x]])/(a^4*d) - b/(2*a^2*d*(a + b*Tan[c + d*x])^2) - (2*b)/(a^3*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 44}"
71,1,178,0,0.1525749,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","-\frac{2 b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right)}{2 a^4 d (a+b \tan (c+d x))^2}-\frac{\left(a^2+6 b^2\right) \cot (c+d x)}{a^5 d}-\frac{b \left(3 a^2+10 b^2\right) \log (\tan (c+d x))}{a^6 d}+\frac{b \left(3 a^2+10 b^2\right) \log (a+b \tan (c+d x))}{a^6 d}+\frac{3 b \cot ^2(c+d x)}{2 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}","-\frac{2 b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right)}{2 a^4 d (a+b \tan (c+d x))^2}-\frac{\left(a^2+6 b^2\right) \cot (c+d x)}{a^5 d}-\frac{b \left(3 a^2+10 b^2\right) \log (\tan (c+d x))}{a^6 d}+\frac{b \left(3 a^2+10 b^2\right) \log (a+b \tan (c+d x))}{a^6 d}+\frac{3 b \cot ^2(c+d x)}{2 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}",1,"-(((a^2 + 6*b^2)*Cot[c + d*x])/(a^5*d)) + (3*b*Cot[c + d*x]^2)/(2*a^4*d) - Cot[c + d*x]^3/(3*a^3*d) - (b*(3*a^2 + 10*b^2)*Log[Tan[c + d*x]])/(a^6*d) + (b*(3*a^2 + 10*b^2)*Log[a + b*Tan[c + d*x]])/(a^6*d) - (b*(a^2 + b^2))/(2*a^4*d*(a + b*Tan[c + d*x])^2) - (2*b*(a^2 + 2*b^2))/(a^5*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
72,1,265,0,0.2394008,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right)^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac{2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^5 d}+\frac{b \left(3 a^2+5 b^2\right) \cot ^2(c+d x)}{a^6 d}-\frac{\left(12 a^2 b^2+a^4+15 b^4\right) \cot (c+d x)}{a^7 d}-\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right) \log (\tan (c+d x))}{a^8 d}+\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right) \log (a+b \tan (c+d x))}{a^8 d}+\frac{3 b \cot ^4(c+d x)}{4 a^4 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}","-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right)^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac{2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^5 d}+\frac{b \left(3 a^2+5 b^2\right) \cot ^2(c+d x)}{a^6 d}-\frac{\left(12 a^2 b^2+a^4+15 b^4\right) \cot (c+d x)}{a^7 d}-\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right) \log (\tan (c+d x))}{a^8 d}+\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right) \log (a+b \tan (c+d x))}{a^8 d}+\frac{3 b \cot ^4(c+d x)}{4 a^4 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}",1,"-(((a^4 + 12*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a^7*d)) + (b*(3*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^6*d) - (2*(a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*a^5*d) + (3*b*Cot[c + d*x]^4)/(4*a^4*d) - Cot[c + d*x]^5/(5*a^3*d) - (b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[Tan[c + d*x]])/(a^8*d) + (b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[a + b*Tan[c + d*x]])/(a^8*d) - (b*(a^2 + b^2)^2)/(2*a^6*d*(a + b*Tan[c + d*x])^2) - (2*b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
73,1,366,0,1.370251,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^4,x]","-\frac{a^4 b}{3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}-\frac{a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{3 a^2 b \left(-5 a^2 b^2+a^4+2 b^4\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \tan (c+d x)+4 a b \left(a^2-b^2\right)\right)}{4 d \left(a^2+b^2\right)^4}-\frac{\cos ^2(c+d x) \left(\left(-65 a^4 b^2+55 a^2 b^4+5 a^6-3 b^6\right) \tan (c+d x)+16 a b \left(-5 a^2 b^2+2 a^4+b^4\right)\right)}{8 d \left(a^2+b^2\right)^5}+\frac{4 a b \left(a^2-b^2\right) \left(-8 a^2 b^2+a^4+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}+\frac{x \left(-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 a^8+3 b^8\right)}{8 \left(a^2+b^2\right)^6}","-\frac{a^4 b}{3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}-\frac{a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{3 a^2 b \left(-5 a^2 b^2+a^4+2 b^4\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \tan (c+d x)+4 a b \left(a^2-b^2\right)\right)}{4 d \left(a^2+b^2\right)^4}-\frac{\cos ^2(c+d x) \left(\left(-65 a^4 b^2+55 a^2 b^4+5 a^6-3 b^6\right) \tan (c+d x)+16 a b \left(-5 a^2 b^2+2 a^4+b^4\right)\right)}{8 d \left(a^2+b^2\right)^5}+\frac{4 a b \left(a^2-b^2\right) \left(-8 a^2 b^2+a^4+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}+\frac{x \left(-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 a^8+3 b^8\right)}{8 \left(a^2+b^2\right)^6}",1,"((3*a^8 - 132*a^6*b^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*x)/(8*(a^2 + b^2)^6) + (4*a*b*(a^2 - b^2)*(a^4 - 8*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^4*b)/(3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3) - (a^3*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^2) - (3*a^2*b*(a^4 - 5*a^2*b^2 + 2*b^4))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(4*(a^2 + b^2)^4*d) - (Cos[c + d*x]^2*(16*a*b*(2*a^4 - 5*a^2*b^2 + b^4) + (5*a^6 - 65*a^4*b^2 + 55*a^2*b^4 - 3*b^6)*Tan[c + d*x]))/(8*(a^2 + b^2)^5*d)","A",8,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
74,1,264,0,0.5708837,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Int[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","-\frac{a^2 b}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^3}-\frac{a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{b \left(-8 a^2 b^2+3 a^4+b^4\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \tan (c+d x)+4 a b \left(a^2-b^2\right)\right)}{2 d \left(a^2+b^2\right)^4}+\frac{4 a b \left(-5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{x \left(-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right)}{2 \left(a^2+b^2\right)^5}","-\frac{a^2 b}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^3}-\frac{a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{b \left(-8 a^2 b^2+3 a^4+b^4\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \tan (c+d x)+4 a b \left(a^2-b^2\right)\right)}{2 d \left(a^2+b^2\right)^4}+\frac{4 a b \left(-5 a^2 b^2+a^4+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{x \left(-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right)}{2 \left(a^2+b^2\right)^5}",1,"((a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(2*(a^2 + b^2)^5) + (4*a*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^2*b)/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^3) - (a*b*(a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 - 8*a^2*b^2 + b^4))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(2*(a^2 + b^2)^4*d)","A",7,6,21,0.2857,1,"{3516, 1647, 1629, 635, 203, 260}"
75,1,116,0,0.0884231,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Int[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","-\frac{3 b}{a^4 d (a+b \tan (c+d x))}-\frac{b}{a^3 d (a+b \tan (c+d x))^2}-\frac{b}{3 a^2 d (a+b \tan (c+d x))^3}-\frac{4 b \log (\tan (c+d x))}{a^5 d}+\frac{4 b \log (a+b \tan (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a^4 d}","-\frac{3 b}{a^4 d (a+b \tan (c+d x))}-\frac{b}{a^3 d (a+b \tan (c+d x))^2}-\frac{b}{3 a^2 d (a+b \tan (c+d x))^3}-\frac{4 b \log (\tan (c+d x))}{a^5 d}+\frac{4 b \log (a+b \tan (c+d x))}{a^5 d}-\frac{\cot (c+d x)}{a^4 d}",1,"-(Cot[c + d*x]/(a^4*d)) - (4*b*Log[Tan[c + d*x]])/(a^5*d) + (4*b*Log[a + b*Tan[c + d*x]])/(a^5*d) - b/(3*a^2*d*(a + b*Tan[c + d*x])^3) - b/(a^3*d*(a + b*Tan[c + d*x])^2) - (3*b)/(a^4*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 44}"
76,1,205,0,0.1732535,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Int[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^4,x]","-\frac{b \left(3 a^2+10 b^2\right)}{a^6 d (a+b \tan (c+d x))}-\frac{b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left(a^2+b^2\right)}{3 a^4 d (a+b \tan (c+d x))^3}-\frac{\left(a^2+10 b^2\right) \cot (c+d x)}{a^6 d}-\frac{4 b \left(a^2+5 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left(a^2+5 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}+\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}","-\frac{b \left(3 a^2+10 b^2\right)}{a^6 d (a+b \tan (c+d x))}-\frac{b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left(a^2+b^2\right)}{3 a^4 d (a+b \tan (c+d x))^3}-\frac{\left(a^2+10 b^2\right) \cot (c+d x)}{a^6 d}-\frac{4 b \left(a^2+5 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left(a^2+5 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}+\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}",1,"-(((a^2 + 10*b^2)*Cot[c + d*x])/(a^6*d)) + (2*b*Cot[c + d*x]^2)/(a^5*d) - Cot[c + d*x]^3/(3*a^4*d) - (4*b*(a^2 + 5*b^2)*Log[Tan[c + d*x]])/(a^7*d) + (4*b*(a^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/(a^7*d) - (b*(a^2 + b^2))/(3*a^4*d*(a + b*Tan[c + d*x])^3) - (b*(a^2 + 2*b^2))/(a^5*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 + 10*b^2))/(a^6*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
77,1,300,0,0.2712431,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]","-\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right)}{a^8 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))^2}-\frac{b \left(a^2+b^2\right)^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{2 \left(a^2+5 b^2\right) \cot ^3(c+d x)}{3 a^6 d}+\frac{2 b \left(2 a^2+5 b^2\right) \cot ^2(c+d x)}{a^7 d}-\frac{\left(20 a^2 b^2+a^4+35 b^4\right) \cot (c+d x)}{a^8 d}-\frac{4 b \left(10 a^2 b^2+a^4+14 b^4\right) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left(10 a^2 b^2+a^4+14 b^4\right) \log (a+b \tan (c+d x))}{a^9 d}+\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}","-\frac{b \left(20 a^2 b^2+3 a^4+21 b^4\right)}{a^8 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))^2}-\frac{b \left(a^2+b^2\right)^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{2 \left(a^2+5 b^2\right) \cot ^3(c+d x)}{3 a^6 d}+\frac{2 b \left(2 a^2+5 b^2\right) \cot ^2(c+d x)}{a^7 d}-\frac{\left(20 a^2 b^2+a^4+35 b^4\right) \cot (c+d x)}{a^8 d}-\frac{4 b \left(10 a^2 b^2+a^4+14 b^4\right) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left(10 a^2 b^2+a^4+14 b^4\right) \log (a+b \tan (c+d x))}{a^9 d}+\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}",1,"-(((a^4 + 20*a^2*b^2 + 35*b^4)*Cot[c + d*x])/(a^8*d)) + (2*b*(2*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^7*d) - (2*(a^2 + 5*b^2)*Cot[c + d*x]^3)/(3*a^6*d) + (b*Cot[c + d*x]^4)/(a^5*d) - Cot[c + d*x]^5/(5*a^4*d) - (4*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[Tan[c + d*x]])/(a^9*d) + (4*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[a + b*Tan[c + d*x]])/(a^9*d) - (b*(a^2 + b^2)^2)/(3*a^6*d*(a + b*Tan[c + d*x])^3) - (b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 + 20*a^2*b^2 + 21*b^4))/(a^8*d*(a + b*Tan[c + d*x]))","A",3,2,21,0.09524,1,"{3516, 894}"
78,1,26,0,0.0744782,"\int \frac{\csc (x)}{1+\tan (x)} \, dx","Int[Csc[x]/(1 + Tan[x]),x]","\frac{\tanh ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right)}{\sqrt{2}}-\tanh ^{-1}(\cos (x))","\frac{\tanh ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right)}{\sqrt{2}}-\tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]] + ArcTanh[(Cos[x] - Sin[x])/Sqrt[2]]/Sqrt[2]","A",6,5,9,0.5556,1,"{3518, 3110, 3770, 3074, 206}"
79,1,229,0,0.4498625,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx","Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^3,x]","\frac{3 a^2 b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}+\frac{b^3 \sin ^{m+4}(c+d x) \, _2F_1\left(2,\frac{m+4}{2};\frac{m+6}{2};\sin ^2(c+d x)\right)}{d (m+4)}","\frac{3 a^2 b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}+\frac{b^3 \sin ^{m+4}(c+d x) \, _2F_1\left(2,\frac{m+4}{2};\frac{m+6}{2};\sin ^2(c+d x)\right)}{d (m+4)}",1,"(a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^2*b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m)) + (3*a*b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(3 + m))/(d*(3 + m)) + (b^3*Hypergeometric2F1[2, (4 + m)/2, (6 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(4 + m))/(d*(4 + m))","A",8,5,21,0.2381,1,"{4401, 2643, 2564, 364, 2577}"
80,1,179,0,0.2680179,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^2 \, dx","Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}","\frac{a^2 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}",1,"(a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a*b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m)) + (b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(3 + m))/(d*(3 + m))","A",6,5,21,0.2381,1,"{4401, 2643, 2564, 364, 2577}"
81,1,109,0,0.1446097,"\int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx","Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x]),x]","\frac{a \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}","\frac{a \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m))","A",5,4,19,0.2105,1,"{4401, 2643, 2564, 364}"
82,1,765,0,4.1731614,"\int \frac{\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx","Int[Sin[c + d*x]^m/(a + b*Tan[c + d*x]),x]","\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)^2}-\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)^2}+\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)}-\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)}+\frac{2^{m+1} \tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{a d (m+1)}","\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)^2}-\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)^2}+\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)}-\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)}+\frac{2^{m+1} \tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{a d (m+1)}",1,"(2^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(a*d*(1 + m)) + (2^(1 + m)*b*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^2*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2])*d*(2 + m)) - (2^(1 + m)*b*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^2*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d*(2 + m)) + (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^3*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2])^2*d*(3 + m)) - (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, (a^2*Tan[(c + d*x)/2]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(c + d*x)/2]^3*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^m*(1 + Tan[(c + d*x)/2]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])^2*d*(3 + m))","A",14,6,21,0.2857,1,"{12, 6719, 6728, 364, 959, 510}"
83,0,0,0,2.4630708,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n,x]","\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin ^m(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Defer[Int][Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
84,1,435,0,0.8030318,"\int \sin ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","-\frac{\left(\sqrt{-b^2} \left(a^2 b^2 \left(-n^2+6 n+6\right)+3 a^4+b^4 \left(n^2+4 n+3\right)\right)+a b^2 n \left(5 a^2+b^2 (2 n+3)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n \left(5 a^2+b^2 (2 n+3)\right)-\sqrt{-b^2} \left(a^2 b^2 \left(-n^2+6 n+6\right)+3 a^4+b^4 \left(n^2+4 n+3\right)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a+\sqrt{-b^2}\right)}+\frac{\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2 (2 n+3)\right) \tan (c+d x)+b \left(a^2 (7-n)+b^2 (n+5)\right)\right) (a+b \tan (c+d x))^{n+1}}{8 d \left(a^2+b^2\right)^2}","-\frac{\left(\sqrt{-b^2} \left(a^2 b^2 \left(-n^2+6 n+6\right)+3 a^4+b^4 \left(n^2+4 n+3\right)\right)+a b^2 n \left(5 a^2+b^2 (2 n+3)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n \left(5 a^2+b^2 (2 n+3)\right)-\sqrt{-b^2} \left(a^2 b^2 \left(-n^2+6 n+6\right)+3 a^4+b^4 \left(n^2+4 n+3\right)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a+\sqrt{-b^2}\right)}+\frac{\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2 (2 n+3)\right) \tan (c+d x)+b \left(a^2 (7-n)+b^2 (n+5)\right)\right) (a+b \tan (c+d x))^{n+1}}{8 d \left(a^2+b^2\right)^2}",1,"-((a*b^2*n*(5*a^2 + b^2*(3 + 2*n)) + Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 + 6*n - n^2) + b^4*(3 + 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*b*(a^2 + b^2)^2*(a - Sqrt[-b^2])*d*(1 + n)) - ((a*b^2*n*(5*a^2 + b^2*(3 + 2*n)) - Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 + 6*n - n^2) + b^4*(3 + 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*b*(a^2 + b^2)^2*(a + Sqrt[-b^2])*d*(1 + n)) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(4*(a^2 + b^2)*d) - (Cos[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)*(b*(a^2*(7 - n) + b^2*(5 + n)) + a*(5*a^2 + b^2*(3 + 2*n))*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)","A",7,4,21,0.1905,1,"{3516, 1649, 831, 68}"
85,1,276,0,0.3673711,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","-\frac{\left(\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)+a b^2 n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n-\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a+\sqrt{-b^2}\right)}-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left(a^2+b^2\right)}","-\frac{\left(\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)+a b^2 n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n-\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a+\sqrt{-b^2}\right)}-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left(a^2+b^2\right)}",1,"-((a*b^2*n + Sqrt[-b^2]*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(a - Sqrt[-b^2])*d*(1 + n)) - ((a*b^2*n - Sqrt[-b^2]*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(a + Sqrt[-b^2])*d*(1 + n)) - (Cos[c + d*x]^2*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(2*(a^2 + b^2)*d)","A",6,4,21,0.1905,1,"{3516, 1649, 831, 68}"
86,1,48,0,0.0548995,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}","\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))","A",2,2,21,0.09524,1,"{3516, 65}"
87,1,140,0,0.1299397,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","\frac{b \left(6 a^2+b^2 \left(n^2-3 n+2\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}+\frac{b (2-n) \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{6 a^2 d}-\frac{\cot ^3(c+d x) (a+b \tan (c+d x))^{n+1}}{3 a d}","\frac{b \left(6 a^2+b^2 \left(n^2-3 n+2\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}+\frac{b (2-n) \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{6 a^2 d}-\frac{\cot ^3(c+d x) (a+b \tan (c+d x))^{n+1}}{3 a d}",1,"(b*(2 - n)*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(6*a^2*d) - (Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(1 + n))/(3*a*d) + (b*(6*a^2 + b^2*(2 - 3*n + n^2))*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(6*a^4*d*(1 + n))","A",4,4,21,0.1905,1,"{3516, 950, 78, 65}"
88,0,0,0,1.9787332,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin ^3(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Defer[Int][Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
89,0,0,0,0.8550524,"\int \sin (c+d x) (a+b \tan (c+d x))^n \, dx","Int[Sin[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\int \sin (c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin (c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Defer[Int][Sin[c + d*x]*(a + b*Tan[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
90,0,0,0,0.4865473,"\int \csc (c+d x) (a+b \tan (c+d x))^n \, dx","Int[Csc[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\int \csc (c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\csc (c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Defer[Int][Csc[c + d*x]*(a + b*Tan[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
91,0,0,0,1.6998061,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\csc ^3(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Defer[Int][Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"