1,1,123,115,0.4470485,"\int (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \sin (c+d x) \tan ^4(c+d x)}{d}-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}-\frac{5 a \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 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,"-((a*Sin[c + d*x]*Tan[c + d*x]^4)/d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (5*a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
2,1,77,71,0.1167245,"\int (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","-\frac{a \sin (c+d x) \tan ^2(c+d x)}{d}+\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}-\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))-\tan (c+d x) \sec (c+d x)\right)}{2 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,"-((a*Sin[c + d*x]*Tan[c + d*x]^2)/d) - (3*a*(ArcTanh[Sin[c + d*x]] - Sec[c + d*x]*Tan[c + d*x]))/(2*d) + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
3,1,38,30,0.0221281,"\int (a+a \sin (c+d x)) \tan (c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x],x]","-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \log (1-\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d - (a*Sin[c + d*x])/d","A",1
4,1,26,24,0.0332121,"\int \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a (\sin (c+d x)+\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]] + Sin[c + d*x]))/d","A",1
5,1,60,54,0.1184625,"\int \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 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*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) - (a*Sin[c + d*x])/d","A",1
6,1,87,81,0.1968147,"\int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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]^3)/(3*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d) + (a*Sin[c + d*x])/d","A",1
7,1,111,115,0.38851,"\int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \left(2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right)}{12 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 + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^5)/(5*d) - (a*(6*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d) - (a*Sin[c + d*x])/d","A",1
8,1,110,101,0.0596952,"\int (a+a \sin (c+d x)) \tan ^6(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^6,x]","\frac{a \cos (c+d x)}{d}-\frac{a \tan ^{-1}(\tan (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}","\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*ArcTan[Tan[c + d*x]])/d) + (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",1
9,1,81,72,0.046453,"\int (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^{-1}(\tan (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}","-\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*ArcTan[Tan[c + d*x]])/d - (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",1
10,1,47,39,0.0386034,"\int (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x",1,"-((a*ArcTan[Tan[c + d*x]])/d) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
11,1,75,41,0.0425583,"\int \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{a \cos (c+d x)}{d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\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*Cos[c + d*x])/d - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d","C",1
12,1,125,82,0.0518026,"\int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\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*Cos[c + d*x])/d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (3*a*Log[Cos[(c + d*x)/2]])/(2*d) - (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","C",1
13,1,164,122,0.0669839,"\int \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}+\frac{a \cos (c+d x)}{d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{9 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{9 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{15 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{15 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}","\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*Cos[c + d*x])/d + (9*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) - (15*a*Log[Cos[(c + d*x)/2]])/(8*d) + (15*a*Log[Sin[(c + d*x)/2]])/(8*d) - (9*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","C",1
14,1,75,119,0.2427338,"\int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{a^2 \left(4 \sin ^2(c+d x)+16 \sin (c+d x)-\frac{18}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+31 \log (1-\sin (c+d x))+\log (\sin (c+d x)+1)\right)}{8 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{a^2 \sin ^2(c+d x)}{2 d}-\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,"-1/8*(a^2*(31*Log[1 - Sin[c + d*x]] + Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 - 18/(-1 + Sin[c + d*x]) + 16*Sin[c + d*x] + 4*Sin[c + d*x]^2))/d","A",1
15,1,54,72,0.1001389,"\int (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \left(\sin ^2(c+d x)+4 \sin (c+d x)+\frac{2}{1-\sin (c+d x)}+6 \log (1-\sin (c+d x))\right)}{2 d}","\frac{a^3}{d (a-a \sin (c+d x))}+\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{3 a^2 \log (1-\sin (c+d x))}{d}",1,"(a^2*(6*Log[1 - Sin[c + d*x]] + 2/(1 - Sin[c + d*x]) + 4*Sin[c + d*x] + Sin[c + d*x]^2))/(2*d)","A",1
16,1,40,52,0.0379066,"\int (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{a^2 \left(\sin ^2(c+d x)+4 \sin (c+d x)+4 \log (1-\sin (c+d x))\right)}{2 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,"-1/2*(a^2*(4*Log[1 - Sin[c + d*x]] + 4*Sin[c + d*x] + Sin[c + d*x]^2))/d","A",1
17,1,28,30,0.0409842,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (\sin (c+d x)+1)^4 \csc ^2(c+d x)}{2 d}","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d}",1,"-1/2*(a^2*Csc[c + d*x]^2*(1 + Sin[c + d*x])^4)/d","A",1
18,1,86,132,0.2203524,"\int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(15 \sin ^2(c+d x)+60 \sin (c+d x)+5 \csc ^6(c+d x)+12 \csc ^5(c+d x)-15 \csc ^4(c+d x)-60 \csc ^3(c+d x)+180 \csc (c+d x)-60 \log (\sin (c+d x))\right)}{30 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,"-1/30*(a^2*(180*Csc[c + d*x] - 60*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 + 12*Csc[c + d*x]^5 + 5*Csc[c + d*x]^6 - 60*Log[Sin[c + d*x]] + 60*Sin[c + d*x] + 15*Sin[c + d*x]^2))/d","A",1
19,1,174,149,0.8393405,"\int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^6,x]","-\frac{a^2 \sec ^5(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 (250 \sin (c+d x)-720 c \sin (2 (c+d x))-720 d x \sin (2 (c+d x))-824 \sin (2 (c+d x))+351 \sin (3 (c+d x))+5 \sin (5 (c+d x))+10 (90 c+90 d x+103) \cos (c+d x)-544 \cos (2 (c+d x))-180 c \cos (3 (c+d x))-180 d x \cos (3 (c+d x))-206 \cos (3 (c+d x))+20 \cos (4 (c+d x))-500)}{160 d}","\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,"-1/160*(a^2*Sec[c + d*x]^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-500 + 10*(103 + 90*c + 90*d*x)*Cos[c + d*x] - 544*Cos[2*(c + d*x)] - 206*Cos[3*(c + d*x)] - 180*c*Cos[3*(c + d*x)] - 180*d*x*Cos[3*(c + d*x)] + 20*Cos[4*(c + d*x)] + 250*Sin[c + d*x] - 824*Sin[2*(c + d*x)] - 720*c*Sin[2*(c + d*x)] - 720*d*x*Sin[2*(c + d*x)] + 351*Sin[3*(c + d*x)] + 5*Sin[5*(c + d*x)]))/d","A",1
20,1,159,120,1.3078014,"\int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{a^2 \left(-21 (12 c+12 d x+7) \cos \left(\frac{1}{2} (c+d x)\right)+(84 c+84 d x+239) \cos \left(\frac{3}{2} (c+d x)\right)+3 \left(-5 \cos \left(\frac{5}{2} (c+d x)\right)+\cos \left(\frac{7}{2} (c+d x)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((28 c+28 d x-27) \cos (c+d x)-6 \cos (2 (c+d x))-\cos (3 (c+d x))+56 c+56 d x+50)\right)\right)}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{16 a^2 \cos (c+d x)}{3 d}-\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,"-1/48*(a^2*(-21*(7 + 12*c + 12*d*x)*Cos[(c + d*x)/2] + (239 + 84*c + 84*d*x)*Cos[(3*(c + d*x))/2] + 3*(-5*Cos[(5*(c + d*x))/2] + Cos[(7*(c + d*x))/2] + 2*(50 + 56*c + 56*d*x + (-27 + 28*c + 28*d*x)*Cos[c + d*x] - 6*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
21,1,145,71,0.4337156,"\int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{a^2 (\sin (c+d x)+1)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right) (10 (c+d x)-\sin (2 (c+d x))-8 \cos (c+d x))+\sin \left(\frac{1}{2} (c+d x)\right) (-2 (5 c+5 d x+8)+\sin (2 (c+d x))+8 \cos (c+d x))\right)}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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,"-1/4*(a^2*(1 + Sin[c + d*x])^2*(Cos[(c + d*x)/2]*(10*(c + d*x) - 8*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*x)/2]*(-2*(8 + 5*c + 5*d*x) + 8*Cos[c + d*x] + Sin[2*(c + d*x)])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","B",1
22,1,34,45,0.1901419,"\int (a+a \sin (c+d x))^2 \, dx","Integrate[(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (-6 (c+d x)+\sin (2 (c+d x))+8 \cos (c+d x))}{4 d}","-\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,"-1/4*(a^2*(-6*(c + d*x) + 8*Cos[c + d*x] + Sin[2*(c + d*x)]))/d","A",1
23,1,94,74,0.5767776,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(7 \cos (c+d x)+\cos (3 (c+d x))+4 \sin (c+d x) \left(-4 \cos (c+d x)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{16 d}","\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,"-1/16*(a^2*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(7*Cos[c + d*x] + Cos[3*(c + d*x)] + 4*(c + d*x - 4*Cos[c + d*x] + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))/d","A",1
24,1,191,98,5.274541,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(-12 (c+d x)-6 \sin (2 (c+d x))-48 \cos (c+d x)-4 \tan \left(\frac{1}{2} (c+d x)\right)+4 \cot \left(\frac{1}{2} (c+d x)\right)-6 \csc ^2\left(\frac{1}{2} (c+d x)\right)+6 \sec ^2\left(\frac{1}{2} (c+d x)\right)-72 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+72 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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*(1 + Sin[c + d*x])^2*(-12*(c + d*x) - 48*Cos[c + d*x] + 4*Cot[(c + d*x)/2] - 6*Csc[(c + d*x)/2]^2 + 72*Log[Cos[(c + d*x)/2]] - 72*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*x)/2]^2 + 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 6*Sin[2*(c + d*x)] - 4*Tan[(c + d*x)/2]))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
25,1,99,160,0.5515593,"\int (a+a \sin (c+d x))^3 \tan ^7(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^7,x]","\frac{a^3 \left(16 \sin ^3(c+d x)+72 \sin ^2(c+d x)+336 \sin (c+d x)-\frac{426}{\sin (c+d x)-1}-\frac{78}{(\sin (c+d x)-1)^2}-\frac{8}{(\sin (c+d x)-1)^3}+627 \log (1-\sin (c+d x))-3 \log (\sin (c+d x)+1)\right)}{48 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{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\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,"(a^3*(627*Log[1 - Sin[c + d*x]] - 3*Log[1 + Sin[c + d*x]] - 8/(-1 + Sin[c + d*x])^3 - 78/(-1 + Sin[c + d*x])^2 - 426/(-1 + Sin[c + d*x]) + 336*Sin[c + d*x] + 72*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3))/(48*d)","A",1
26,1,66,91,0.1579502,"\int (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{a^3 \left(2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+30 \sin (c+d x)+\frac{12}{1-\sin (c+d x)}+42 \log (1-\sin (c+d x))\right)}{6 d}","\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{5 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \log (1-\sin (c+d x))}{d}",1,"(a^3*(42*Log[1 - Sin[c + d*x]] + 12/(1 - Sin[c + d*x]) + 30*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/(6*d)","A",1
27,1,52,70,0.0449727,"\int (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{a^3 \left(2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+24 \sin (c+d x)+24 \log (1-\sin (c+d x))\right)}{6 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,"-1/6*(a^3*(24*Log[1 - Sin[c + d*x]] + 24*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/d","A",1
28,1,67,98,0.1864081,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+12 \sin (c+d x)+3 \csc ^2(c+d x)+18 \csc (c+d x)-12 \log (\sin (c+d x))+30\right)}{6 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,"-1/6*(a^3*(30 + 18*Csc[c + d*x] + 3*Csc[c + d*x]^2 - 12*Log[Sin[c + d*x]] + 12*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/d","A",1
29,1,243,180,5.1166668,"\int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^6,x]","\frac{(a \sin (c+d x)+a)^3 \left(-690 (c+d x)+45 \sin (2 (c+d x))+405 \cos (c+d x)-5 \cos (3 (c+d x))+\frac{1576 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{224 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{24 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}-\frac{112}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{12}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}\right)}{60 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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{136 a^3 \cos ^3(c+d x)}{15 d}+\frac{136 a^3 \cos (c+d x)}{5 d}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{23 a^3 x}{2}+\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)}",1,"((a + a*Sin[c + d*x])^3*(-690*(c + d*x) + 405*Cos[c + d*x] - 5*Cos[3*(c + d*x)] + 12/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 - 112/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (24*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 - (224*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (1576*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 45*Sin[2*(c + d*x)]))/(60*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
30,1,177,119,2.104044,"\int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{(a \sin (c+d x)+a)^3 \left(102 (c+d x)-9 \sin (2 (c+d x))-69 \cos (c+d x)+\cos (3 (c+d x))-\frac{200 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{8}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"((a + a*Sin[c + d*x])^3*(102*(c + d*x) - 69*Cos[c + d*x] + Cos[3*(c + d*x)] + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (200*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
31,1,115,89,0.4829649,"\int (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{(a \sin (c+d x)+a)^3 \left(-66 (c+d x)+9 \sin (2 (c+d x))+57 \cos (c+d x)-\cos (3 (c+d x))+\frac{96 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"((a + a*Sin[c + d*x])^3*(-66*(c + d*x) + 57*Cos[c + d*x] - Cos[3*(c + d*x)] + (96*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
32,1,44,63,0.3152506,"\int (a+a \sin (c+d x))^3 \, dx","Integrate[(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (-9 \sin (2 (c+d x))-45 \cos (c+d x)+\cos (3 (c+d x))+30 c+30 d x)}{12 d}","\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,"(a^3*(30*c + 30*d*x - 45*Cos[c + d*x] + Cos[3*(c + d*x)] - 9*Sin[2*(c + d*x)]))/(12*d)","A",1
33,1,106,92,1.0882281,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((15-66 \sin (c+d x)) \cos (c+d x)+(2 \sin (c+d x)+9) \cos (3 (c+d x))-12 \sin (c+d x) \left(6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{48 d}","-\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,"-1/48*(a^3*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Cos[c + d*x]*(15 - 66*Sin[c + d*x]) - 12*(c + d*x - 6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]])*Sin[c + d*x] + Cos[3*(c + d*x)]*(9 + 2*Sin[c + d*x])))/d","A",1
34,1,83,129,0.446097,"\int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^5,x]","-\frac{a^4 \left(3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+54 \sin ^2(c+d x)+192 \sin (c+d x)+\frac{120-132 \sin (c+d x)}{(\sin (c+d x)-1)^2}+300 \log (1-\sin (c+d x))\right)}{12 d}","\frac{a^6}{d (a-a \sin (c+d x))^2}-\frac{11 a^5}{d (a-a \sin (c+d 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{16 a^4 \sin (c+d x)}{d}-\frac{25 a^4 \log (1-\sin (c+d x))}{d}",1,"-1/12*(a^4*(300*Log[1 - Sin[c + d*x]] + (120 - 132*Sin[c + d*x])/(-1 + Sin[c + d*x])^2 + 192*Sin[c + d*x] + 54*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/d","A",1
35,1,76,107,0.1512025,"\int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]","\frac{a^4 \left(3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+48 \sin ^2(c+d x)+144 \sin (c+d x)+\frac{48}{1-\sin (c+d x)}+192 \log (1-\sin (c+d x))\right)}{12 d}","\frac{4 a^5}{d (a-a \sin (c+d 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{12 a^4 \sin (c+d x)}{d}+\frac{16 a^4 \log (1-\sin (c+d x))}{d}",1,"(a^4*(192*Log[1 - Sin[c + d*x]] + 48/(1 - Sin[c + d*x]) + 144*Sin[c + d*x] + 48*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/(12*d)","A",1
36,1,62,88,0.0685275,"\int (a+a \sin (c+d x))^4 \tan (c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x],x]","-\frac{a^4 \left(3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+42 \sin ^2(c+d x)+96 \sin (c+d x)+96 \log (1-\sin (c+d x))\right)}{12 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,"-1/12*(a^4*(96*Log[1 - Sin[c + d*x]] + 96*Sin[c + d*x] + 42*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/d","A",1
37,1,78,102,0.1300235,"\int \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 \sin ^4(c+d x) \left(6 \csc ^6(c+d x)+48 \csc ^5(c+d x)+30 \csc ^2(c+d x)+16 \csc (c+d x)+\csc ^4(c+d x) (90-60 \log (\sin (c+d x)))+3\right)}{12 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,"-1/12*(a^4*(3 + 16*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 48*Csc[c + d*x]^5 + 6*Csc[c + d*x]^6 + Csc[c + d*x]^4*(90 - 60*Log[Sin[c + d*x]]))*Sin[c + d*x]^4)/d","A",1
38,1,252,143,1.6525217,"\int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]","\frac{a^4 \left(-11736 c \sin \left(\frac{1}{2} (c+d x)\right)-11736 d x \sin \left(\frac{1}{2} (c+d x)\right)-16488 \sin \left(\frac{1}{2} (c+d x)\right)-3912 c \sin \left(\frac{3}{2} (c+d x)\right)-3912 d x \sin \left(\frac{3}{2} (c+d x)\right)+3704 \sin \left(\frac{3}{2} (c+d x)\right)+885 \sin \left(\frac{5}{2} (c+d x)\right)+129 \sin \left(\frac{7}{2} (c+d x)\right)-23 \sin \left(\frac{9}{2} (c+d x)\right)-3 \sin \left(\frac{11}{2} (c+d x)\right)+24 (489 c+489 d x+209) \cos \left(\frac{1}{2} (c+d x)\right)-24 (163 c+163 d x+453) \cos \left(\frac{3}{2} (c+d x)\right)+885 \cos \left(\frac{5}{2} (c+d x)\right)-129 \cos \left(\frac{7}{2} (c+d x)\right)-23 \cos \left(\frac{9}{2} (c+d x)\right)+3 \cos \left(\frac{11}{2} (c+d x)\right)\right)}{384 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(a^4*(24*(209 + 489*c + 489*d*x)*Cos[(c + d*x)/2] - 24*(453 + 163*c + 163*d*x)*Cos[(3*(c + d*x))/2] + 885*Cos[(5*(c + d*x))/2] - 129*Cos[(7*(c + d*x))/2] - 23*Cos[(9*(c + d*x))/2] + 3*Cos[(11*(c + d*x))/2] - 16488*Sin[(c + d*x)/2] - 11736*c*Sin[(c + d*x)/2] - 11736*d*x*Sin[(c + d*x)/2] + 3704*Sin[(3*(c + d*x))/2] - 3912*c*Sin[(3*(c + d*x))/2] - 3912*d*x*Sin[(3*(c + d*x))/2] + 885*Sin[(5*(c + d*x))/2] + 129*Sin[(7*(c + d*x))/2] - 23*Sin[(9*(c + d*x))/2] - 3*Sin[(11*(c + d*x))/2]))/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
39,1,125,113,1.1007125,"\int (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^2,x]","\frac{(a \sin (c+d x)+a)^4 \left(-1140 (c+d x)+192 \sin (2 (c+d x))-3 \sin (4 (c+d x))+1056 \cos (c+d x)-32 \cos (3 (c+d x))+\frac{1536 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{96 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"((a + a*Sin[c + d*x])^4*(-1140*(c + d*x) + 1056*Cos[c + d*x] - 32*Cos[3*(c + d*x)] + (1536*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 192*Sin[2*(c + d*x)] - 3*Sin[4*(c + d*x)]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)","A",1
40,1,57,87,0.3986431,"\int (a+a \sin (c+d x))^4 \, dx","Integrate[(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (3 (-56 \sin (2 (c+d x))+\sin (4 (c+d x))+140 c+140 d x)-672 \cos (c+d x)+32 \cos (3 (c+d x)))}{96 d}","\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,"(a^4*(-672*Cos[c + d*x] + 32*Cos[3*(c + d*x)] + 3*(140*c + 140*d*x - 56*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(96*d)","A",1
41,1,136,116,1.5980201,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(408 c \sin (c+d x)+408 d x \sin (c+d x)+320 \sin (2 (c+d x))-32 \sin (4 (c+d x))-48 \cos (c+d x)-147 \cos (3 (c+d x))+3 \cos (5 (c+d x))+768 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-768 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d}","-\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,"(a^4*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(-48*Cos[c + d*x] - 147*Cos[3*(c + d*x)] + 3*Cos[5*(c + d*x)] + 408*c*Sin[c + d*x] + 408*d*x*Sin[c + d*x] - 768*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 768*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 320*Sin[2*(c + d*x)] - 32*Sin[4*(c + d*x)]))/(384*d)","A",1
42,1,209,140,5.2794246,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (\sin (c+d x)+1)^4 \left(-732 (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+96 \cos (c+d x)+32 \cos (3 (c+d x))+224 \tan \left(\frac{1}{2} (c+d x)\right)-224 \cot \left(\frac{1}{2} (c+d x)\right)-48 \csc ^2\left(\frac{1}{2} (c+d x)\right)+48 \sec ^2\left(\frac{1}{2} (c+d x)\right)-192 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+192 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+32 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{96 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"(a^4*(1 + Sin[c + d*x])^4*(-732*(c + d*x) + 96*Cos[c + d*x] + 32*Cos[3*(c + d*x)] - 224*Cot[(c + d*x)/2] - 48*Csc[(c + d*x)/2]^2 + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 48*Sec[(c + d*x)/2]^2 + 32*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 120*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] + 224*Tan[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)","A",1
43,1,283,198,1.5782218,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (\sin (c+d x)+1)^4 \left(5820 (c+d x)+480 \sin (2 (c+d x))-15 \sin (4 (c+d x))-2400 \cos (c+d x)-160 \cos (3 (c+d x))-2752 \tan \left(\frac{1}{2} (c+d x)\right)+2752 \cot \left(\frac{1}{2} (c+d x)\right)-30 \csc ^4\left(\frac{1}{2} (c+d x)\right)+300 \csc ^2\left(\frac{1}{2} (c+d x)\right)+30 \sec ^4\left(\frac{1}{2} (c+d x)\right)-300 \sec ^2\left(\frac{1}{2} (c+d x)\right)-1200 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1200 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{3}{2} \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+96 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-\frac{79}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+632 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{480 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"(a^4*(1 + Sin[c + d*x])^4*(5820*(c + d*x) - 2400*Cos[c + d*x] - 160*Cos[3*(c + d*x)] + 2752*Cot[(c + d*x)/2] + 300*Csc[(c + d*x)/2]^2 - 30*Csc[(c + d*x)/2]^4 + 1200*Log[Cos[(c + d*x)/2]] - 1200*Log[Sin[(c + d*x)/2]] - 300*Sec[(c + d*x)/2]^2 + 30*Sec[(c + d*x)/2]^4 + 632*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - (79*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - (3*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 480*Sin[2*(c + d*x)] - 15*Sin[4*(c + d*x)] - 2752*Tan[(c + d*x)/2]))/(480*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)","A",1
44,1,101,130,0.9827209,"\int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\frac{279 \sin ^6(c+d x)+87 \sin ^5(c+d x)-424 \sin ^4(c+d x)-136 \sin ^3(c+d x)+249 \sin ^2(c+d x)+57 \sin (c+d x)-48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+105 \tanh ^{-1}(\sin (c+d x))}{384 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,"-1/384*(105*ArcTanh[Sin[c + d*x]] + (-48 + 57*Sin[c + d*x] + 249*Sin[c + d*x]^2 - 136*Sin[c + d*x]^3 - 424*Sin[c + d*x]^4 + 87*Sin[c + d*x]^5 + 279*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(a*d)","A",1
45,1,84,106,0.325537,"\int \frac{\tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sin[c + d*x]),x]","\frac{-\frac{18}{1-\sin (c+d x)}+\frac{48}{\sin (c+d x)+1}+\frac{3}{(1-\sin (c+d x))^2}-\frac{21}{(\sin (c+d x)+1)^2}+\frac{4}{(\sin (c+d x)+1)^3}+30 \tanh ^{-1}(\sin (c+d x))}{96 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,"(30*ArcTanh[Sin[c + d*x]] + 3/(1 - Sin[c + d*x])^2 - 18/(1 - Sin[c + d*x]) + 4/(1 + Sin[c + d*x])^3 - 21/(1 + Sin[c + d*x])^2 + 48/(1 + Sin[c + d*x]))/(96*a*d)","A",1
46,1,54,82,0.1641392,"\int \frac{\tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{\frac{1}{\sin (c+d x)-1}+\frac{4}{\sin (c+d x)+1}-\frac{1}{(\sin (c+d x)+1)^2}+3 \tanh ^{-1}(\sin (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,"-1/8*(3*ArcTanh[Sin[c + d*x]] + (-1 + Sin[c + d*x])^(-1) - (1 + Sin[c + d*x])^(-2) + 4/(1 + Sin[c + d*x]))/(a*d)","A",1
47,1,28,37,0.0356312,"\int \frac{\tan (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\frac{1}{\sin (c+d x)+1}+\tanh ^{-1}(\sin (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]] + (1 + Sin[c + d*x])^(-1))/(2*a*d)","A",1
48,1,32,32,0.018216,"\int \frac{\cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[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",1
49,1,24,32,0.0303807,"\int \frac{\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{(\csc (c+d x)-2) \csc (c+d x)}{2 a d}","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"-1/2*((-2 + Csc[c + d*x])*Csc[c + d*x])/(a*d)","A",1
50,1,30,51,0.0460985,"\int \frac{\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x]),x]","-\frac{(\csc (c+d x)-1)^3 (3 \csc (c+d x)+5)}{12 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,"-1/12*((-1 + Csc[c + d*x])^3*(5 + 3*Csc[c + d*x]))/(a*d)","A",1
51,1,61,68,0.14122,"\int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]","\frac{\csc ^6(c+d x) (78 \sin (c+d x)-5 (7 \sin (3 (c+d x))-3 \sin (5 (c+d x))+5)-15 \cos (4 (c+d x)))}{240 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,"(Csc[c + d*x]^6*(-15*Cos[4*(c + d*x)] + 78*Sin[c + d*x] - 5*(5 + 7*Sin[3*(c + d*x)] - 3*Sin[5*(c + d*x)])))/(240*a*d)","A",1
52,1,77,84,0.2061362,"\int \frac{\cot ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x]),x]","\frac{\csc ^8(c+d x) (-513 \sin (c+d x)+371 \sin (3 (c+d x))-105 \sin (5 (c+d x))+35 \sin (7 (c+d x))-245 \cos (2 (c+d x))-35 \cos (6 (c+d x)))}{2240 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,"(Csc[c + d*x]^8*(-245*Cos[2*(c + d*x)] - 35*Cos[6*(c + d*x)] - 513*Sin[c + d*x] + 371*Sin[3*(c + d*x)] - 105*Sin[5*(c + d*x)] + 35*Sin[7*(c + d*x)]))/(2240*a*d)","A",1
53,1,146,84,0.3149879,"\int \frac{\tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + a*Sin[c + d*x]),x]","\frac{\sec ^5(c+d x) (2432 \sin (c+d x)-1905 \sin (2 (c+d x))+320 \sin (3 (c+d x))-1524 \sin (4 (c+d x))+960 \sin (5 (c+d x))-381 \sin (6 (c+d x))-7620 \cos (c+d x)+3760 \cos (2 (c+d x))-3810 \cos (3 (c+d x))+1440 \cos (4 (c+d x))-762 \cos (5 (c+d x))+80 \cos (6 (c+d x))+2912)}{17920 a d (\sin (c+d x)+1)}","\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]^5*(2912 - 7620*Cos[c + d*x] + 3760*Cos[2*(c + d*x)] - 3810*Cos[3*(c + d*x)] + 1440*Cos[4*(c + d*x)] - 762*Cos[5*(c + d*x)] + 80*Cos[6*(c + d*x)] + 2432*Sin[c + d*x] - 1905*Sin[2*(c + d*x)] + 320*Sin[3*(c + d*x)] - 1524*Sin[4*(c + d*x)] + 960*Sin[5*(c + d*x)] - 381*Sin[6*(c + d*x)]))/(17920*a*d*(1 + Sin[c + d*x]))","A",1
54,1,106,69,0.3248304,"\int \frac{\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) (-64 \sin (c+d x)-178 \sin (2 (c+d x))+192 \sin (3 (c+d x))-89 \sin (4 (c+d x))-534 \cos (c+d x)+288 \cos (2 (c+d x))-178 \cos (3 (c+d x))+24 \cos (4 (c+d x))+200)}{960 a d (\sin (c+d x)+1)}","\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,"-1/960*(Sec[c + d*x]^3*(200 - 534*Cos[c + d*x] + 288*Cos[2*(c + d*x)] - 178*Cos[3*(c + d*x)] + 24*Cos[4*(c + d*x)] - 64*Sin[c + d*x] - 178*Sin[2*(c + d*x)] + 192*Sin[3*(c + d*x)] - 89*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
55,1,106,50,0.1616402,"\int \frac{\tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{8 \sin (c+d x)-5 \sin (2 (c+d x))-10 \cos (c+d x)+2 \cos (2 (c+d x))+6}{12 a d (\sin (c+d x)+1) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(6 - 10*Cos[c + d*x] + 2*Cos[2*(c + d*x)] + 8*Sin[c + d*x] - 5*Sin[2*(c + d*x)])/(12*a*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))","B",1
56,1,48,23,0.0413186,"\int \frac{1}{a+a \sin (c+d x)} \, dx","Integrate[(a + a*Sin[c + d*x])^(-1),x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \sin (c+d x)+a)}","-\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}",1,"(2*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*(a + a*Sin[c + d*x]))","B",1
57,1,69,29,0.2313669,"\int \frac{\cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\cos (c+d x)+\sin (c+d x) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2 a d}","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}",1,"-1/2*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Cos[c + d*x] + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))/(a*d)","B",1
58,1,124,58,0.5090598,"\int \frac{\cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\cos (3 (c+d x))+(3-6 \sin (c+d x)) \cos (c+d x)+6 \sin ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{96 a d (\sin (c+d x)+1)}","-\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,"-1/96*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(Cos[3*(c + d*x)] + Cos[c + d*x]*(3 - 6*Sin[c + d*x]) + 6*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3))/(a*d*(1 + Sin[c + d*x]))","B",1
59,1,189,82,0.7593541,"\int \frac{\cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(20 \sin (2 (c+d x))-50 \sin (4 (c+d x))+80 \cos (c+d x)+40 \cos (3 (c+d x))+8 \cos (5 (c+d x))+150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{640 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,"-1/640*(Csc[c + d*x]^5*(80*Cos[c + d*x] + 40*Cos[3*(c + d*x)] + 8*Cos[5*(c + d*x)] - 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 20*Sin[2*(c + d*x)] + 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 50*Sin[4*(c + d*x)] - 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a*d)","B",1
60,1,284,106,0.9244474,"\int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-1190 \sin (2 (c+d x))+392 \sin (4 (c+d x))-462 \sin (6 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-3675 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2205 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-735 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3675 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2205 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+735 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{86016 a d (\sin (c+d x)+1)}","-\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,"-1/86016*(Csc[c + d*x]^5*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d*x)] + 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 1190*Sin[2*(c + d*x)] - 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 392*Sin[4*(c + d*x)] + 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 462*Sin[6*(c + d*x)] - 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
61,1,112,189,1.6404165,"\int \frac{\tan ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]","-\frac{210 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left(105 \sin ^7(c+d x)-750 \sin ^6(c+d x)-815 \sin ^5(c+d x)+560 \sin ^4(c+d x)+1039 \sin ^3(c+d x)+78 \sin ^2(c+d x)-393 \sin (c+d x)-144\right)}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^5}}{3840 a^2 d}","\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,"-1/3840*(210*ArcTanh[Sin[c + d*x]] - (2*(-144 - 393*Sin[c + d*x] + 78*Sin[c + d*x]^2 + 1039*Sin[c + d*x]^3 + 560*Sin[c + d*x]^4 - 815*Sin[c + d*x]^5 - 750*Sin[c + d*x]^6 + 105*Sin[c + d*x]^7))/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^5))/(a^2*d)","A",1
62,1,91,146,0.4471245,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{-15 \sin ^5(c+d x)+66 \sin ^4(c+d x)+74 \sin ^3(c+d x)-14 \sin ^2(c+d x)-47 \sin (c+d x)-16}{(\sin (c+d x)-1)^2 (\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{192 a^2 d}","\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,"(15*ArcTanh[Sin[c + d*x]] + (-16 - 47*Sin[c + d*x] - 14*Sin[c + d*x]^2 + 74*Sin[c + d*x]^3 + 66*Sin[c + d*x]^4 - 15*Sin[c + d*x]^5)/((-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^4))/(192*a^2*d)","A",1
63,1,70,104,0.3174141,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","-\frac{-\frac{3}{1-\sin (c+d x)}-\frac{9}{\sin (c+d x)+1}+\frac{12}{(\sin (c+d x)+1)^2}-\frac{4}{(\sin (c+d x)+1)^3}+6 \tanh ^{-1}(\sin (c+d x))}{48 a^2 d}","\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,"-1/48*(6*ArcTanh[Sin[c + d*x]] - 3/(1 - Sin[c + d*x]) - 4/(1 + Sin[c + d*x])^3 + 12/(1 + Sin[c + d*x])^2 - 9/(1 + Sin[c + d*x]))/(a^2*d)","A",1
64,1,36,60,0.0812422,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{\tanh ^{-1}(\sin (c+d x))-\frac{\sin (c+d x)}{(\sin (c+d x)+1)^2}}{4 a^2 d}","-\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]] - Sin[c + d*x]/(1 + Sin[c + d*x])^2)/(4*a^2*d)","A",1
65,1,36,52,0.0567445,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{1}{\sin (c+d x)+1}+\log (\sin (c+d x))-\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]] - Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-1))/(a^2*d)","A",1
66,1,49,65,0.0680928,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","\frac{-\csc ^2(c+d x)+4 \csc (c+d x)+4 \log (\sin (c+d x))-4 \log (\sin (c+d x)+1)}{2 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,"(4*Csc[c + d*x] - Csc[c + d*x]^2 + 4*Log[Sin[c + d*x]] - 4*Log[1 + Sin[c + d*x]])/(2*a^2*d)","A",1
67,1,38,55,0.0666074,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^4(c+d x) (8 \sin (c+d x)+3 \cos (2 (c+d x))-6)}{12 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]^4*(-6 + 3*Cos[2*(c + d*x)] + 8*Sin[c + d*x]))/(12*a^2*d)","A",1
68,1,73,73,0.0736676,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[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",1
69,1,78,127,0.1511652,"\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^2(c+d x) \left(-105 \csc ^6(c+d x)+240 \csc ^5(c+d x)+140 \csc ^4(c+d x)-672 \csc ^3(c+d x)+210 \csc ^2(c+d x)+560 \csc (c+d x)-420\right)}{840 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*(-420 + 560*Csc[c + d*x] + 210*Csc[c + d*x]^2 - 672*Csc[c + d*x]^3 + 140*Csc[c + d*x]^4 + 240*Csc[c + d*x]^5 - 105*Csc[c + d*x]^6))/(840*a^2*d)","A",1
70,1,88,145,0.2144732,"\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^2(c+d x) \left(-126 \csc ^8(c+d x)+280 \csc ^7(c+d x)+315 \csc ^6(c+d x)-1080 \csc ^5(c+d x)+1512 \csc ^3(c+d x)-630 \csc ^2(c+d x)-840 \csc (c+d x)+630\right)}{1260 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*(630 - 840*Csc[c + d*x] - 630*Csc[c + d*x]^2 + 1512*Csc[c + d*x]^3 - 1080*Csc[c + d*x]^5 + 315*Csc[c + d*x]^6 + 280*Csc[c + d*x]^7 - 126*Csc[c + d*x]^8))/(1260*a^2*d)","A",1
71,1,118,199,0.3416228,"\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x) \left(1155 \csc ^{10}(c+d x)-2520 \csc ^9(c+d x)-4158 \csc ^8(c+d x)+12320 \csc ^7(c+d x)+3465 \csc ^6(c+d x)-23760 \csc ^5(c+d x)+4620 \csc ^4(c+d x)+22176 \csc ^3(c+d x)-10395 \csc ^2(c+d x)-9240 \csc (c+d x)+6930\right)}{13860 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,"-1/13860*(Csc[c + d*x]^2*(6930 - 9240*Csc[c + d*x] - 10395*Csc[c + d*x]^2 + 22176*Csc[c + d*x]^3 + 4620*Csc[c + d*x]^4 - 23760*Csc[c + d*x]^5 + 3465*Csc[c + d*x]^6 + 12320*Csc[c + d*x]^7 - 4158*Csc[c + d*x]^8 - 2520*Csc[c + d*x]^9 + 1155*Csc[c + d*x]^10))/(a^2*d)","A",1
72,1,102,171,0.6758121,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{15 \tanh ^{-1}(\sin (c+d x))-\frac{15 \sin ^6(c+d x)+45 \sin ^5(c+d x)-620 \sin ^4(c+d x)-540 \sin ^3(c+d x)+157 \sin ^2(c+d x)+351 \sin (c+d x)+112}{(\sin (c+d x)-1)^2 (\sin (c+d x)+1)^5}}{1920 a^3 d}","-\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{a^2}{40 d (a \sin (c+d x)+a)^5}-\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,"(15*ArcTanh[Sin[c + d*x]] - (112 + 351*Sin[c + d*x] + 157*Sin[c + d*x]^2 - 540*Sin[c + d*x]^3 - 620*Sin[c + d*x]^4 + 45*Sin[c + d*x]^5 + 15*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^5))/(1920*a^3*d)","A",1
73,1,82,126,0.3607099,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","-\frac{-\frac{3}{1-\sin (c+d x)}-\frac{6}{\sin (c+d x)+1}-\frac{9}{(\sin (c+d x)+1)^2}+\frac{16}{(\sin (c+d x)+1)^3}-\frac{6}{(\sin (c+d x)+1)^4}+3 \tanh ^{-1}(\sin (c+d x))}{96 a^3 d}","\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,"-1/96*(3*ArcTanh[Sin[c + d*x]] - 3/(1 - Sin[c + d*x]) - 6/(1 + Sin[c + d*x])^4 + 16/(1 + Sin[c + d*x])^3 - 9/(1 + Sin[c + d*x])^2 - 6/(1 + Sin[c + d*x]))/(a^3*d)","A",1
74,1,52,82,0.1507802,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{3 \tanh ^{-1}(\sin (c+d x))-\frac{3 \sin ^2(c+d x)+9 \sin (c+d x)+2}{(\sin (c+d x)+1)^3}}{24 a^3 d}","-\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,"(3*ArcTanh[Sin[c + d*x]] - (2 + 9*Sin[c + d*x] + 3*Sin[c + d*x]^2)/(1 + Sin[c + d*x])^3)/(24*a^3*d)","A",1
75,1,52,74,0.1817359,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x)+3}{(\sin (c+d x)+1)^2}+2 \log (\sin (c+d x))-2 \log (\sin (c+d x)+1)}{2 a^3 d}","\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,"(2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (3 + 2*Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(2*a^3*d)","A",1
76,1,61,86,0.1910501,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{4}{\sin (c+d x)+1}-\csc ^2(c+d x)+6 \csc (c+d x)+10 \log (\sin (c+d x))-10 \log (\sin (c+d x)+1)}{2 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,"(6*Csc[c + d*x] - Csc[c + d*x]^2 + 10*Log[Sin[c + d*x]] - 10*Log[1 + Sin[c + d*x]] + 4/(1 + Sin[c + d*x]))/(2*a^3*d)","A",1
77,1,69,96,0.31438,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{-\csc ^4(c+d x)+4 \csc ^3(c+d x)-8 \csc ^2(c+d x)+16 \csc (c+d x)+16 \log (\sin (c+d x))-16 \log (\sin (c+d x)+1)}{4 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,"(16*Csc[c + d*x] - 8*Csc[c + d*x]^2 + 4*Csc[c + d*x]^3 - Csc[c + d*x]^4 + 16*Log[Sin[c + d*x]] - 16*Log[1 + Sin[c + d*x]])/(4*a^3*d)","A",1
78,1,48,73,0.097406,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^3(c+d x) \left(-10 \csc ^3(c+d x)+36 \csc ^2(c+d x)-45 \csc (c+d x)+20\right)}{60 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*(20 - 45*Csc[c + d*x] + 36*Csc[c + d*x]^2 - 10*Csc[c + d*x]^3))/(60*a^3*d)","A",1
79,1,68,109,0.0757174,"\int \frac{\cot ^9(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^3(c+d x) \left(105 \csc ^5(c+d x)-360 \csc ^4(c+d x)+280 \csc ^3(c+d x)+336 \csc ^2(c+d x)-630 \csc (c+d x)+280\right)}{840 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,"-1/840*(Csc[c + d*x]^3*(280 - 630*Csc[c + d*x] + 336*Csc[c + d*x]^2 + 280*Csc[c + d*x]^3 - 360*Csc[c + d*x]^4 + 105*Csc[c + d*x]^5))/(a^3*d)","A",1
80,1,88,145,0.1114796,"\int \frac{\cot ^{11}(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^3(c+d x) \left(-84 \csc ^7(c+d x)+280 \csc ^6(c+d x)-105 \csc ^5(c+d x)-600 \csc ^4(c+d x)+700 \csc ^3(c+d x)+168 \csc ^2(c+d x)-630 \csc (c+d x)+280\right)}{840 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*(280 - 630*Csc[c + d*x] + 168*Csc[c + d*x]^2 + 700*Csc[c + d*x]^3 - 600*Csc[c + d*x]^4 - 105*Csc[c + d*x]^5 + 280*Csc[c + d*x]^6 - 84*Csc[c + d*x]^7))/(840*a^3*d)","A",1
81,1,88,145,0.1197157,"\int \frac{\cot ^{13}(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^3(c+d x) \left(-231 \csc ^9(c+d x)+756 \csc ^8(c+d x)-2464 \csc ^6(c+d x)+2079 \csc ^5(c+d x)+2376 \csc ^4(c+d x)-3696 \csc ^3(c+d x)+2079 \csc (c+d x)-924\right)}{2772 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*(-924 + 2079*Csc[c + d*x] - 3696*Csc[c + d*x]^3 + 2376*Csc[c + d*x]^4 + 2079*Csc[c + d*x]^5 - 2464*Csc[c + d*x]^6 + 756*Csc[c + d*x]^8 - 231*Csc[c + d*x]^9))/(2772*a^3*d)","A",1
82,1,112,195,1.4463108,"\int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]","-\frac{30 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left(15 \sin ^7(c+d x)+60 \sin ^6(c+d x)+65 \sin ^5(c+d x)+440 \sin ^4(c+d x)+257 \sin ^3(c+d x)-132 \sin ^2(c+d x)-177 \sin (c+d x)-48\right)}{(\sin (c+d x)-1)^2 (\sin (c+d x)+1)^6}}{3840 a^4 d}","-\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{\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}+\frac{a^2}{48 d (a \sin (c+d x)+a)^6}+\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{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,"-1/3840*(30*ArcTanh[Sin[c + d*x]] - (2*(-48 - 177*Sin[c + d*x] - 132*Sin[c + d*x]^2 + 257*Sin[c + d*x]^3 + 440*Sin[c + d*x]^4 + 65*Sin[c + d*x]^5 + 60*Sin[c + d*x]^6 + 15*Sin[c + d*x]^7))/((-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^6))/(a^4*d)","A",1
83,1,50,132,0.1044388,"\int \frac{\tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^3/(a + a*Sin[c + d*x])^4,x]","-\frac{5 \sin ^2(c+d x)+4 \sin (c+d x)+1}{20 a^4 d (\sin (c+d x)-1) (\sin (c+d x)+1)^5}","\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,"-1/20*(1 + 4*Sin[c + d*x] + 5*Sin[c + d*x]^2)/(a^4*d*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^5)","A",1
84,1,62,105,0.256415,"\int \frac{\tan (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Tan[c + d*x]/(a + a*Sin[c + d*x])^4,x]","\frac{3 \tanh ^{-1}(\sin (c+d x))-\frac{3 \sin ^3(c+d x)+12 \sin ^2(c+d x)+19 \sin (c+d x)+4}{(\sin (c+d x)+1)^4}}{48 a^4 d}","-\frac{1}{16 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{16 a^4 d}-\frac{1}{16 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{1}{12 a d (a \sin (c+d x)+a)^3}+\frac{1}{8 d (a \sin (c+d x)+a)^4}",1,"(3*ArcTanh[Sin[c + d*x]] - (4 + 19*Sin[c + d*x] + 12*Sin[c + d*x]^2 + 3*Sin[c + d*x]^3)/(1 + Sin[c + d*x])^4)/(48*a^4*d)","A",1
85,1,73,106,0.8018536,"\int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x])^4,x]","\frac{\frac{10}{\sin (c+d x)+1}+\frac{2}{(\sin (c+d x)+1)^2}-\csc ^2(c+d x)+8 \csc (c+d x)+18 \log (\sin (c+d x))-18 \log (\sin (c+d x)+1)}{2 a^4 d}","\frac{5}{d \left(a^4 \sin (c+d x)+a^4\right)}-\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{1}{d \left(a^2 \sin (c+d x)+a^2\right)^2}",1,"(8*Csc[c + d*x] - Csc[c + d*x]^2 + 18*Log[Sin[c + d*x]] - 18*Log[1 + Sin[c + d*x]] + 2/(1 + Sin[c + d*x])^2 + 10/(1 + Sin[c + d*x]))/(2*a^4*d)","A",1
86,1,89,135,0.16098,"\int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]","\frac{-10 \csc ^6(c+d x)+48 \csc ^5(c+d x)-105 \csc ^4(c+d x)+160 \csc ^3(c+d x)-240 \csc ^2(c+d x)+480 \csc (c+d x)+480 \log (\sin (c+d x))-480 \log (\sin (c+d x)+1)}{60 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,"(480*Csc[c + d*x] - 240*Csc[c + d*x]^2 + 160*Csc[c + d*x]^3 - 105*Csc[c + d*x]^4 + 48*Csc[c + d*x]^5 - 10*Csc[c + d*x]^6 + 480*Log[Sin[c + d*x]] - 480*Log[1 + Sin[c + d*x]])/(60*a^4*d)","A",1
87,1,124,127,0.4284727,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^4,x]","\frac{\sec (c+d x) (34944 \sin (c+d x)+1776 \sin (2 (c+d x))-9504 \sin (3 (c+d x))-296 \sin (4 (c+d x))+352 \sin (5 (c+d x))+1554 \cos (c+d x)-16896 \cos (2 (c+d x))-999 \cos (3 (c+d x))+2816 \cos (4 (c+d x))+37 \cos (5 (c+d x))+16128)}{80640 a^4 d (\sin (c+d x)+1)^4}","\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,"(Sec[c + d*x]*(16128 + 1554*Cos[c + d*x] - 16896*Cos[2*(c + d*x)] - 999*Cos[3*(c + d*x)] + 2816*Cos[4*(c + d*x)] + 37*Cos[5*(c + d*x)] + 34944*Sin[c + d*x] + 1776*Sin[2*(c + d*x)] - 9504*Sin[3*(c + d*x)] - 296*Sin[4*(c + d*x)] + 352*Sin[5*(c + d*x)]))/(80640*a^4*d*(1 + Sin[c + d*x])^4)","A",1
88,1,315,108,0.4232392,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^4,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(24 \sin \left(\frac{1}{2} (c+d x)\right)+316 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-38 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+76 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-12 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5-120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5+15 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5-15 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5\right)}{30 d (a \sin (c+d x)+a)^4}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(24*Sin[(c + d*x)/2] - 12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 76*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 38*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 316*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 15*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + 120*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - 120*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + 15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*Tan[(c + d*x)/2]))/(30*d*(a + a*Sin[c + d*x])^4)","B",1
89,1,589,120,6.0880789,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]","\frac{80 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}{3 d (a \sin (c+d x)+a)^4}-\frac{4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{3 d (a \sin (c+d x)+a)^4}+\frac{8 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}{3 d (a \sin (c+d x)+a)^4}+\frac{14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{d (a \sin (c+d x)+a)^4}-\frac{14 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{d (a \sin (c+d x)+a)^4}+\frac{13 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{3 d (a \sin (c+d x)+a)^4}-\frac{13 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{3 d (a \sin (c+d x)+a)^4}+\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{2 d (a \sin (c+d x)+a)^4}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{2 d (a \sin (c+d x)+a)^4}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{24 d (a \sin (c+d x)+a)^4}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{24 d (a \sin (c+d x)+a)^4}","-\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,"(8*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(3*d*(a + a*Sin[c + d*x])^4) - (4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(3*d*(a + a*Sin[c + d*x])^4) + (80*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)/(3*d*(a + a*Sin[c + d*x])^4) - (13*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(3*d*(a + a*Sin[c + d*x])^4) + (Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(24*d*(a + a*Sin[c + d*x])^4) + (14*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(d*(a + a*Sin[c + d*x])^4) - (14*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(d*(a + a*Sin[c + d*x])^4) - (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) + (13*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(3*d*(a + a*Sin[c + d*x])^4) + (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(24*d*(a + a*Sin[c + d*x])^4)","B",1
90,1,733,133,6.1321113,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]","\frac{16 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}{d (a \sin (c+d x)+a)^4}+\frac{27 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{2 d (a \sin (c+d x)+a)^4}-\frac{27 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{2 d (a \sin (c+d x)+a)^4}+\frac{33 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{5 d (a \sin (c+d x)+a)^4}-\frac{33 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{5 d (a \sin (c+d x)+a)^4}+\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{16 d (a \sin (c+d x)+a)^4}+\frac{11 \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{8 d (a \sin (c+d x)+a)^4}-\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{16 d (a \sin (c+d x)+a)^4}-\frac{11 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{8 d (a \sin (c+d x)+a)^4}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{160 d (a \sin (c+d x)+a)^4}-\frac{53 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{160 d (a \sin (c+d x)+a)^4}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{160 d (a \sin (c+d x)+a)^4}+\frac{53 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}{160 d (a \sin (c+d x)+a)^4}","-\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,"(16*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)/(d*(a + a*Sin[c + d*x])^4) - (33*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(5*d*(a + a*Sin[c + d*x])^4) + (11*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x])^4) - (53*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^4) + (Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(16*d*(a + a*Sin[c + d*x])^4) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^4) + (27*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) - (27*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) - (11*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x])^4) - (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(16*d*(a + a*Sin[c + d*x])^4) + (33*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(5*d*(a + a*Sin[c + d*x])^4) + (53*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(160*d*(a + a*Sin[c + d*x])^4) + (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(160*d*(a + a*Sin[c + d*x])^4)","B",1
91,1,394,162,5.5679484,"\int \sqrt{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*Tan[e + f*x]^4,x]","\frac{\sqrt{a (\sin (e+f x)+1)} \left(-48 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \cos \left(\frac{f x}{2}\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+48 \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \sin \left(\frac{f x}{2}\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-\frac{36 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{3 \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)}+\frac{6 \sin \left(\frac{f x}{2}\right)}{\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)}+(33+33 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{f x}{4}\right) \left(\cos \left(\frac{1}{4} (2 e+f x)\right)-\sin \left(\frac{1}{4} (2 e+f x)\right)\right)\right)\right)}{24 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"(((6*Sin[(f*x)/2])/(Cos[e/2] + Sin[e/2]) - (3*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(Cos[e/2] + Sin[e/2]) + (33 + 33*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(f*x)/4]*(Cos[(2*e + f*x)/4] - Sin[(2*e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 48*Cos[(f*x)/2]*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 48*(Cos[e/2] + Sin[e/2])*Sin[(f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - (36*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))*Sqrt[a*(1 + Sin[e + f*x])])/(24*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","C",1
92,1,114,101,0.3364956,"\int \sqrt{a+a \sin (e+f x)} \tan ^2(e+f x) \, dx","Integrate[Sqrt[a + a*Sin[e + f*x]]*Tan[e + f*x]^2,x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \left(-2 \sin (e+f x)+(1-i) \sqrt[4]{-1} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{f x}{4}\right) \left(\cos \left(\frac{1}{4} (2 e+f x)\right)-\sin \left(\frac{1}{4} (2 e+f x)\right)\right)\right)+3\right)}{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,"(Sec[e + f*x]*(3 + (1 - I)*(-1)^(1/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(f*x)/4]*(Cos[(2*e + f*x)/4] - Sin[(2*e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 2*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])])/f","C",1
93,1,206,89,0.9939113,"\int \cot ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]],x]","\frac{\csc ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(4 \sin \left(\frac{1}{2} (e+f x)\right)+2 \sin \left(\frac{3}{2} (e+f x)\right)-4 \cos \left(\frac{1}{2} (e+f x)\right)+2 \cos \left(\frac{3}{2} (e+f x)\right)-\sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+\sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)-\sec \left(\frac{1}{4} (e+f x)\right)\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)+\sec \left(\frac{1}{4} (e+f x)\right)\right)}","\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,"(Csc[(e + f*x)/2]^4*Sqrt[a*(1 + Sin[e + f*x])]*(-4*Cos[(e + f*x)/2] + 2*Cos[(3*(e + f*x))/2] + 4*Sin[(e + f*x)/2] - Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 2*Sin[(3*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4] - Sec[(e + f*x)/4])*(Csc[(e + f*x)/4] + Sec[(e + f*x)/4]))","B",1
94,1,309,163,1.6034388,"\int \cot ^4(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]","\frac{\csc ^{10}\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(-252 \sin \left(\frac{1}{2} (e+f x)\right)-250 \sin \left(\frac{3}{2} (e+f x)\right)+114 \sin \left(\frac{5}{2} (e+f x)\right)+48 \sin \left(\frac{7}{2} (e+f x)\right)+252 \cos \left(\frac{1}{2} (e+f x)\right)-250 \cos \left(\frac{3}{2} (e+f x)\right)-114 \cos \left(\frac{5}{2} (e+f x)\right)+48 \cos \left(\frac{7}{2} (e+f x)\right)+99 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-99 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-33 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+33 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{24 f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}","-\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,"(Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(252*Cos[(e + f*x)/2] - 250*Cos[(3*(e + f*x))/2] - 114*Cos[(5*(e + f*x))/2] + 48*Cos[(7*(e + f*x))/2] - 252*Sin[(e + f*x)/2] + 99*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 99*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 250*Sin[(3*(e + f*x))/2] + 114*Sin[(5*(e + f*x))/2] - 33*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 33*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 48*Sin[(7*(e + f*x))/2]))/(24*f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)","A",1
95,1,141,167,5.5469281,"\int (a+a \sin (e+f x))^{3/2} \tan ^4(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^4,x]","\frac{a \sec ^3(e+f x) \sqrt{a (\sin (e+f x)+1)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(54 \sin (e+f x)+\sin (3 (e+f x))+6 \cos (2 (e+f x))+(3+3 i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-45\right)}{6 f}","-\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^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{\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*Sec[e + f*x]^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[a*(1 + Sin[e + f*x])]*(-45 + 6*Cos[2*(e + f*x)] + (3 + 3*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 54*Sin[e + f*x] + Sin[3*(e + f*x)]))/(6*f)","C",1
96,1,46,88,4.1518234,"\int (a+a \sin (e+f x))^{3/2} \tan ^2(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^2,x]","\frac{a \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} (-8 \sin (e+f x)+\cos (2 (e+f x))+15)}{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,"(a*Sec[e + f*x]*(15 + Cos[2*(e + f*x)] - 8*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])])/(3*f)","A",1
97,1,233,121,0.7560739,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{a \csc ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(-14 \sin \left(\frac{1}{2} (e+f x)\right)-9 \sin \left(\frac{3}{2} (e+f x)\right)-\sin \left(\frac{5}{2} (e+f x)\right)+14 \cos \left(\frac{1}{2} (e+f x)\right)-9 \cos \left(\frac{3}{2} (e+f x)\right)+\cos \left(\frac{5}{2} (e+f x)\right)+9 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-9 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{3 f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)-\sec \left(\frac{1}{4} (e+f x)\right)\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)+\sec \left(\frac{1}{4} (e+f x)\right)\right)}","-\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{11 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}+\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,"-1/3*(a*Csc[(e + f*x)/2]^4*Sqrt[a*(1 + Sin[e + f*x])]*(14*Cos[(e + f*x)/2] - 9*Cos[(3*(e + f*x))/2] + Cos[(5*(e + f*x))/2] - 14*Sin[(e + f*x)/2] + 9*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 9*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 9*Sin[(3*(e + f*x))/2] - Sin[(5*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4] - Sec[(e + f*x)/4])*(Csc[(e + f*x)/4] + Sec[(e + f*x)/4]))","A",1
98,1,334,197,1.6281038,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{a \csc ^{10}\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(276 \sin \left(\frac{1}{2} (e+f x)\right)+326 \sin \left(\frac{3}{2} (e+f x)\right)-78 \sin \left(\frac{5}{2} (e+f x)\right)-72 \sin \left(\frac{7}{2} (e+f x)\right)-8 \sin \left(\frac{9}{2} (e+f x)\right)-276 \cos \left(\frac{1}{2} (e+f x)\right)+326 \cos \left(\frac{3}{2} (e+f x)\right)+78 \cos \left(\frac{5}{2} (e+f x)\right)-72 \cos \left(\frac{7}{2} (e+f x)\right)+8 \cos \left(\frac{9}{2} (e+f x)\right)-333 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+333 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+111 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-111 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{24 f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}","\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{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{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,"-1/24*(a*Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(-276*Cos[(e + f*x)/2] + 326*Cos[(3*(e + f*x))/2] + 78*Cos[(5*(e + f*x))/2] - 72*Cos[(7*(e + f*x))/2] + 8*Cos[(9*(e + f*x))/2] + 276*Sin[(e + f*x)/2] - 333*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 333*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 326*Sin[(3*(e + f*x))/2] - 78*Sin[(5*(e + f*x))/2] + 111*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 111*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 72*Sin[(7*(e + f*x))/2] - 8*Sin[(9*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)","A",1
99,1,112,151,5.4713299,"\int (a+a \sin (e+f x))^{5/2} \tan ^4(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*Tan[e + f*x]^4,x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} (1488 \sin (e+f x)+16 \sin (3 (e+f x))+204 \cos (2 (e+f x))-3 \cos (4 (e+f x))-1225)}{60 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"(a^2*Sqrt[a*(1 + Sin[e + f*x])]*(-1225 + 204*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)] + 1488*Sin[e + f*x] + 16*Sin[3*(e + f*x)]))/(60*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
100,1,60,118,5.4733229,"\int (a+a \sin (e+f x))^{5/2} \tan ^2(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(5/2)*Tan[e + f*x]^2,x]","\frac{a^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} (-185 \sin (e+f x)+3 \sin (3 (e+f x))+22 \cos (2 (e+f x))+330)}{30 f}","\frac{124 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{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,"(a^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(330 + 22*Cos[2*(e + f*x)] - 185*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))/(30*f)","A",1
101,1,261,151,1.2609895,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^{5/2} \, dx","Integrate[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2),x]","-\frac{a^2 \csc ^4\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(-125 \sin \left(\frac{1}{2} (e+f x)\right)-93 \sin \left(\frac{3}{2} (e+f x)\right)-25 \sin \left(\frac{5}{2} (e+f x)\right)+3 \sin \left(\frac{7}{2} (e+f x)\right)+125 \cos \left(\frac{1}{2} (e+f x)\right)-93 \cos \left(\frac{3}{2} (e+f x)\right)+25 \cos \left(\frac{5}{2} (e+f x)\right)+3 \cos \left(\frac{7}{2} (e+f x)\right)+150 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-150 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{30 f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)-\sec \left(\frac{1}{4} (e+f x)\right)\right) \left(\csc \left(\frac{1}{4} (e+f x)\right)+\sec \left(\frac{1}{4} (e+f x)\right)\right)}","-\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{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{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,"-1/30*(a^2*Csc[(e + f*x)/2]^4*Sqrt[a*(1 + Sin[e + f*x])]*(125*Cos[(e + f*x)/2] - 93*Cos[(3*(e + f*x))/2] + 25*Cos[(5*(e + f*x))/2] + 3*Cos[(7*(e + f*x))/2] - 125*Sin[(e + f*x)/2] + 150*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 150*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 93*Sin[(3*(e + f*x))/2] - 25*Sin[(5*(e + f*x))/2] + 3*Sin[(7*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4] - Sec[(e + f*x)/4])*(Csc[(e + f*x)/4] + Sec[(e + f*x)/4]))","A",1
102,1,360,227,1.7443548,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx","Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]","-\frac{a^2 \csc ^{10}\left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sin (e+f x)+1)} \left(-108 \sin \left(\frac{1}{2} (e+f x)\right)+706 \sin \left(\frac{3}{2} (e+f x)\right)+450 \sin \left(\frac{5}{2} (e+f x)\right)-156 \sin \left(\frac{7}{2} (e+f x)\right)-100 \sin \left(\frac{9}{2} (e+f x)\right)+12 \sin \left(\frac{11}{2} (e+f x)\right)+108 \cos \left(\frac{1}{2} (e+f x)\right)+706 \cos \left(\frac{3}{2} (e+f x)\right)-450 \cos \left(\frac{5}{2} (e+f x)\right)-156 \cos \left(\frac{7}{2} (e+f x)\right)+100 \cos \left(\frac{9}{2} (e+f x)\right)+12 \cos \left(\frac{11}{2} (e+f x)\right)-2475 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+2475 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+825 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-825 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{120 f \left(\cot \left(\frac{1}{2} (e+f x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}","\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{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{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,"-1/120*(a^2*Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(108*Cos[(e + f*x)/2] + 706*Cos[(3*(e + f*x))/2] - 450*Cos[(5*(e + f*x))/2] - 156*Cos[(7*(e + f*x))/2] + 100*Cos[(9*(e + f*x))/2] + 12*Cos[(11*(e + f*x))/2] - 108*Sin[(e + f*x)/2] - 2475*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 2475*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 706*Sin[(3*(e + f*x))/2] + 450*Sin[(5*(e + f*x))/2] + 825*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 825*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 156*Sin[(7*(e + f*x))/2] - 100*Sin[(9*(e + f*x))/2] + 12*Sin[(11*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)","A",1
103,1,118,150,0.6888287,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]],x]","\frac{-\sec ^3(e+f x) (-41 \sin (e+f x)+183 \sin (3 (e+f x))+122 \cos (2 (e+f x))+90)+(804+804 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)}{768 f \sqrt{a (\sin (e+f x)+1)}}","\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,"((804 + 804*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - Sec[e + f*x]^3*(90 + 122*Cos[2*(e + f*x)] - 41*Sin[e + f*x] + 183*Sin[3*(e + f*x)]))/(768*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
104,1,118,107,0.2563631,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\sec (e+f x) \left(-3 \sin (e+f x)+(5+5 i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-1\right)}{4 f \sqrt{a (\sin (e+f x)+1)}}","\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,"-1/4*(Sec[e + f*x]*(-1 + (5 + 5*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 3*Sin[e + f*x]))/(f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
105,1,138,62,0.3214591,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right) \csc \left(\frac{1}{4} (e+f x)\right) \sec \left(\frac{1}{4} (e+f x)\right) \left(2 \sin \left(\frac{1}{2} (e+f x)\right)-2 \cos \left(\frac{1}{2} (e+f x)\right)+\sin (e+f x) \left(\log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)\right)}{8 f \sqrt{a (\sin (e+f x)+1)}}","\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,"(Csc[(e + f*x)/4]*Sec[(e + f*x)/4]*(-2*Cos[(e + f*x)/2] + 2*Sin[(e + f*x)/2] + (Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x])*(1 + Tan[(e + f*x)/2]))/(8*f*Sqrt[a*(1 + Sin[e + f*x])])","B",1
106,1,292,135,0.6078935,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\csc ^9\left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-36 \sin \left(\frac{1}{2} (e+f x)\right)-46 \sin \left(\frac{3}{2} (e+f x)\right)+54 \sin \left(\frac{5}{2} (e+f x)\right)+36 \cos \left(\frac{1}{2} (e+f x)\right)-46 \cos \left(\frac{3}{2} (e+f x)\right)-54 \cos \left(\frac{5}{2} (e+f x)\right)-63 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+63 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+21 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-21 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{24 f \sqrt{a (\sin (e+f x)+1)} \left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}","\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,"(Csc[(e + f*x)/2]^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(36*Cos[(e + f*x)/2] - 46*Cos[(3*(e + f*x))/2] - 54*Cos[(5*(e + f*x))/2] - 36*Sin[(e + f*x)/2] - 63*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 63*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 46*Sin[(3*(e + f*x))/2] + 54*Sin[(5*(e + f*x))/2] + 21*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 21*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(24*f*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3*Sqrt[a*(1 + Sin[e + f*x])])","B",1
107,1,334,177,0.3729843,"\int \frac{\tan ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2),x]","\frac{-\frac{192 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{32 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-171 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+342 \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\frac{248 \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-\frac{32}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+(-21-21 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+124}{768 f (a (\sin (e+f x)+1))^{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,"(124 + (64*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 32/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (248*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 342*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 171*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (21 + 21*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + (32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - (192*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/(768*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
108,1,128,134,0.4562873,"\int \frac{\tan ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\sec (e+f x) \left(-40 \sin (e+f x)-\cos (2 (e+f x))+(2+2 i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-25\right)}{64 f (a (\sin (e+f x)+1))^{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,"-1/64*(Sec[e + f*x]*(-25 - Cos[2*(e + f*x)] + (2 + 2*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 40*Sin[e + f*x]))/(f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
109,1,206,113,2.1146588,"\int \frac{\cot ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(-\cot \left(\frac{1}{4} (e+f x)\right)+(16+16 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+2 \left(\sec \left(\frac{1}{2} (e+f x)\right)+\sin ^2\left(\frac{1}{4} (e+f x)\right) \csc (e+f x)-\sin \left(\frac{3}{4} (e+f x)\right) \sin \left(\frac{1}{4} (e+f x)\right) \csc (e+f x)+3 \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)\right)}{4 f (a (\sin (e+f x)+1))^{3/2}}","\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,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*((16 + 16*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - Cot[(e + f*x)/4] + 2*(3*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 3*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sec[(e + f*x)/2] + Csc[e + f*x]*Sin[(e + f*x)/4]^2 - Csc[e + f*x]*Sin[(e + f*x)/4]*Sin[(3*(e + f*x))/4])))/(4*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
110,1,294,144,0.7641027,"\int \frac{\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\csc ^9\left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(132 \sin \left(\frac{1}{2} (e+f x)\right)+62 \sin \left(\frac{3}{2} (e+f x)\right)-6 \sin \left(\frac{5}{2} (e+f x)\right)-132 \cos \left(\frac{1}{2} (e+f x)\right)+62 \cos \left(\frac{3}{2} (e+f x)\right)+6 \cos \left(\frac{5}{2} (e+f x)\right)-9 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+9 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+3 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-3 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{24 f (a (\sin (e+f x)+1))^{3/2} \left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}","-\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,"(Csc[(e + f*x)/2]^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-132*Cos[(e + f*x)/2] + 62*Cos[(3*(e + f*x))/2] + 6*Cos[(5*(e + f*x))/2] + 132*Sin[(e + f*x)/2] - 9*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 9*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 62*Sin[(3*(e + f*x))/2] - 6*Sin[(5*(e + f*x))/2] + 3*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 3*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(24*f*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3*(a*(1 + Sin[e + f*x]))^(3/2))","B",0
111,1,394,207,0.540919,"\int \frac{\tan ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]","\frac{-\frac{1152 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{256 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-201 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4+402 \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3-1292 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+2584 \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\frac{2624 \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-\frac{384}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{768 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+(-951-951 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)+1312}{12288 f (a (\sin (e+f x)+1))^{5/2}}","\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,"(1312 + (768*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 384/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - (2624*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2584*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 1292*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 402*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 201*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - (951 + 951*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + (256*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 - (1152*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/(12288*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
112,1,284,167,0.4019724,"\int \frac{\tan ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\frac{48 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+15 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4-30 \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3+52 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-104 \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}+(33+33 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)-32}{384 f (a (\sin (e+f x)+1))^{5/2}}","-\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 \sec (e+f x)}{96 a^2 f \sqrt{a \sin (e+f x)+a}}-\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,"(-32 + (64*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 104*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 52*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 30*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 15*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (33 + 33*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + (48*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))/(384*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
113,1,451,141,0.7395815,"\int \frac{\cot ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(8 \sin \left(\frac{1}{2} (e+f x)\right)+\frac{2 \sin \left(\frac{1}{4} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}{\cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)}-\frac{2 \sin \left(\frac{1}{4} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}{\sin \left(\frac{1}{4} (e+f x)\right)+\cos \left(\frac{1}{4} (e+f x)\right)}+2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-4 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+10 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-10 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-\tan \left(\frac{1}{4} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2-\cot \left(\frac{1}{4} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+(28+28 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{4 f (a (\sin (e+f x)+1))^{5/2}}","\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,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(8*Sin[(e + f*x)/2] - 4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (28 + 28*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - Cot[(e + f*x)/4]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 10*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 10*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (2*Sin[(e + f*x)/4]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(Cos[(e + f*x)/4] - Sin[(e + f*x)/4]) - (2*Sin[(e + f*x)/4]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(Cos[(e + f*x)/4] + Sin[(e + f*x)/4]) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Tan[(e + f*x)/4]))/(4*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",1
114,1,332,191,2.4113234,"\int \frac{\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(-\frac{8 \csc ^9\left(\frac{1}{2} (e+f x)\right) \left(-396 \sin \left(\frac{1}{2} (e+f x)\right)-218 \sin \left(\frac{3}{2} (e+f x)\right)+114 \sin \left(\frac{5}{2} (e+f x)\right)+396 \cos \left(\frac{1}{2} (e+f x)\right)-218 \cos \left(\frac{3}{2} (e+f x)\right)-114 \cos \left(\frac{5}{2} (e+f x)\right)-405 \sin (e+f x) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+405 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)+135 \sin (3 (e+f x)) \log \left(-\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+1\right)-135 \sin (3 (e+f x)) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)+1\right)\right)}{\left(\csc ^2\left(\frac{1}{4} (e+f x)\right)-\sec ^2\left(\frac{1}{4} (e+f x)\right)\right)^3}+(1536+1536 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{192 f (a (\sin (e+f x)+1))^{5/2}}","\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{19 \cot (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a}}-\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,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*((1536 + 1536*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] - (8*Csc[(e + f*x)/2]^9*(396*Cos[(e + f*x)/2] - 218*Cos[(3*(e + f*x))/2] - 114*Cos[(5*(e + f*x))/2] - 396*Sin[(e + f*x)/2] - 405*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 405*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 218*Sin[(3*(e + f*x))/2] + 114*Sin[(5*(e + f*x))/2] + 135*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 135*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3))/(192*f*(a*(1 + Sin[e + f*x]))^(5/2))","C",0
115,1,318,982,3.2367563,"\int \sqrt[3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(1/3)*Tan[e + f*x]^4,x]","\frac{\sqrt[3]{a (\sin (e+f x)+1)} \left(3 \left(-172 \tan (e+f x)-3 \sec ^3(e+f x)+86 \sec (e+f x)+24 \tan (e+f x) \sec ^2(e+f x)+361\right)+\frac{\left(\frac{1083}{10}+\frac{1083 i}{10}\right) (-1)^{3/4} e^{-i (e+f x)} \left(-2 \left(1+i e^{-i (e+f x)}\right)^{2/3} \left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (e+f x)}\right) \sqrt{2-2 \sin (e+f x)}+20 e^{i (e+f x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (e+f x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)}\right)}{\sqrt{2} \left(1+i e^{-i (e+f x)}\right)^{2/3} \sqrt{i e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2}}\right)}{189 f}","-\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,"((a*(1 + Sin[e + f*x]))^(1/3)*(((1083/10 + (1083*I)/10)*(-1)^(3/4)*(20*E^(I*(e + f*x))*Sqrt[Cos[(2*e + Pi + 2*f*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(e + f*x))] - 2*(1 + I/E^(I*(e + f*x)))^(2/3)*(1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*e + Pi + 2*f*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(e + f*x))]*Sqrt[2 - 2*Sin[e + f*x]]))/(Sqrt[2]*E^(I*(e + f*x))*(1 + I/E^(I*(e + f*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]) + 3*(361 + 86*Sec[e + f*x] - 3*Sec[e + f*x]^3 - 172*Tan[e + f*x] + 24*Sec[e + f*x]^2*Tan[e + f*x])))/(189*f)","C",0
116,1,290,123,2.7089004,"\int \sqrt[3]{a+a \sin (e+f x)} \tan ^2(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^(1/3)*Tan[e + f*x]^2,x]","\frac{\sqrt[3]{a (\sin (e+f x)+1)} \left(-3 (-2 \tan (e+f x)+\sec (e+f x)+5)+\frac{\left(\frac{3}{2}+\frac{3 i}{2}\right) (-1)^{3/4} e^{-i (e+f x)} \left(2 \left(1+i e^{-i (e+f x)}\right)^{2/3} \left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)-5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (e+f x)}\right) \sqrt{2-2 \sin (e+f x)}-20 e^{i (e+f x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (e+f x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)}\right)}{\sqrt{2} \left(1+i e^{-i (e+f x)}\right)^{2/3} \sqrt{i e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2}}\right)}{3 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,"((a*(1 + Sin[e + f*x]))^(1/3)*(((3/2 + (3*I)/2)*(-1)^(3/4)*(-20*E^(I*(e + f*x))*Sqrt[Cos[(2*e + Pi + 2*f*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(e + f*x))] + 2*(1 + I/E^(I*(e + f*x)))^(2/3)*(1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*e + Pi + 2*f*x)/4]^2] - (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(e + f*x))]*Sqrt[2 - 2*Sin[e + f*x]]))/(Sqrt[2]*E^(I*(e + f*x))*(1 + I/E^(I*(e + f*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]) - 3*(5 + Sec[e + f*x] - 2*Tan[e + f*x])))/(3*f)","C",1
117,1,2692,80,23.7103547,"\int \cot ^2(e+f x) \sqrt[3]{a+a \sin (e+f x)} \, dx","Integrate[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(1/3),x]","\text{Result too large to show}","\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,"((15/2 + (15*I)/2)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*(a*(1 + Sin[e + f*x]))^(1/3))/(f*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])] + (AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])] + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])])*(1 + Cot[(e + f*x)/2]))) + ((-4 - Cot[e + f*x])*(a*(1 + Sin[e + f*x]))^(1/3))/f + ((5/2 + (5*I)/2)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])]*(a*(1 + Sin[e + f*x]))^(1/3))/(f*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])] + (AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])] + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])])*(1 + Tan[(e + f*x)/2]))) + (Cos[(3*(e + f*x))/2]*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(a*(1 + Sin[e + f*x]))^(1/3)*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2])))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(1 + Tan[(e + f*x)/2])*((-3*Sec[(e + f*x)/2]^2*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2])))/(4*(1 + Tan[(e + f*x)/2])^2) + ((8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2]))*(Sqrt[Sec[(e + f*x)/2]^2]/2 - (Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]))/(2*Sqrt[Sec[(e + f*x)/2]^2])))/((1 + Tan[(e + f*x)/2])*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(1/3)) + (3*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(-1/2*(AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*Sec[(e + f*x)/2]^2) + ((1 + I)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*Sec[(e + f*x)/2]^2)/2^(1/3) + ((1/3 + I/3)*2^(2/3)*(((1/2 - I/2)*(I + Cot[(e + f*x)/2])*Csc[(e + f*x)/2]^2)/(1 + Cot[(e + f*x)/2])^2 - ((1/2 - I/2)*Csc[(e + f*x)/2]^2)/(1 + Cot[(e + f*x)/2]))*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]))/(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(2/3) - ((1/6 + I/6)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/((-1 - I)*(I + Cot[(e + f*x)/2]))^(2/3) - ((1/3 - I/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(2/3) - ((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*((-1/30 + I/30)*AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*Csc[(e + f*x)/2]^2 - (1/30 + I/30)*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*Csc[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]) + ((2/3 + (2*I)/3)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*(I + Tan[(e + f*x)/2])*(2 + 2*Tan[(e + f*x)/2])*(-((Sec[(e + f*x)/2]^2*((1 + I) + (1 - I)*Tan[(e + f*x)/2]))/(2 + 2*Tan[(e + f*x)/2])^2) + ((1/2 - I/2)*Sec[(e + f*x)/2]^2)/(2 + 2*Tan[(e + f*x)/2]))*(-Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])] + (1 - ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2]))^(-1/3)))/((1 + I) + (1 - I)*Tan[(e + f*x)/2])))/(2*(1 + Tan[(e + f*x)/2]))))","C",0
118,1,2796,80,22.3346171,"\int \cot ^4(e+f x) \sqrt[3]{a+a \sin (e+f x)} \, dx","Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(1/3),x]","\text{Result too large to show}","\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,"((239/54 + (77*Cot[e + f*x])/54 - (Cot[e + f*x]*Csc[e + f*x])/18 - (Cot[e + f*x]*Csc[e + f*x]^2)/3)*(a*(1 + Sin[e + f*x]))^(1/3))/f - ((70/9 + (70*I)/9)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*(a*(1 + Sin[e + f*x]))^(1/3)*(1 + Tan[(e + f*x)/2]))/(f*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*Sec[(e + f*x)/2] + AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*(Csc[(e + f*x)/2] + Sec[(e + f*x)/2]) + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*(Csc[(e + f*x)/2] + Sec[(e + f*x)/2]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - ((355/108 + (355*I)/108)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])]*(a*(1 + Sin[e + f*x]))^(1/3))/(f*((5 + 5*I)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])] + (AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])] + I*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Tan[(e + f*x)/2]), (1/2 - I/2)*(1 + Tan[(e + f*x)/2])])*(1 + Tan[(e + f*x)/2]))) - (239*Cos[(3*(e + f*x))/2]*Csc[e + f*x]*(a*(1 + Sin[e + f*x]))^(1/3)*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2])))/(216*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(1 + Tan[(e + f*x)/2])*((-3*Sec[(e + f*x)/2]^2*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2])))/(8*(1 + Tan[(e + f*x)/2])^2) + ((8 + (1 + I)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]) - AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*(1 + Tan[(e + f*x)/2]))*(Sqrt[Sec[(e + f*x)/2]^2]/2 - (Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]))/(2*Sqrt[Sec[(e + f*x)/2]^2])))/(2*(1 + Tan[(e + f*x)/2])*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(1/3)) + (3*((1 + Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2/3)*(-1/2*(AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*Sec[(e + f*x)/2]^2) + ((1 + I)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*Sec[(e + f*x)/2]^2)/2^(1/3) + ((1/3 + I/3)*2^(2/3)*(((1/2 - I/2)*(I + Cot[(e + f*x)/2])*Csc[(e + f*x)/2]^2)/(1 + Cot[(e + f*x)/2])^2 - ((1/2 - I/2)*Csc[(e + f*x)/2]^2)/(1 + Cot[(e + f*x)/2]))*Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])]*(I + Tan[(e + f*x)/2]))/(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(2/3) - ((1/6 + I/6)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/((-1 - I)*(I + Cot[(e + f*x)/2]))^(2/3) - ((1/3 - I/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(2/3) - ((2 + 2*I) - (2 - 2*I)*Cot[(e + f*x)/2])^(1/3)*((-1 - I)*(I + Cot[(e + f*x)/2]))^(1/3)*((-1/30 + I/30)*AppellF1[5/3, 1/3, 4/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*Csc[(e + f*x)/2]^2 - (1/30 + I/30)*AppellF1[5/3, 4/3, 1/3, 8/3, (1/2 + I/2)*(1 + Cot[(e + f*x)/2]), (1/2 - I/2)*(1 + Cot[(e + f*x)/2])]*Csc[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]) + ((2/3 + (2*I)/3)*2^(2/3)*(((1 - I)*(I + Cot[(e + f*x)/2]))/(1 + Cot[(e + f*x)/2]))^(1/3)*(I + Tan[(e + f*x)/2])*(2 + 2*Tan[(e + f*x)/2])*(-((Sec[(e + f*x)/2]^2*((1 + I) + (1 - I)*Tan[(e + f*x)/2]))/(2 + 2*Tan[(e + f*x)/2])^2) + ((1/2 - I/2)*Sec[(e + f*x)/2]^2)/(2 + 2*Tan[(e + f*x)/2]))*(-Hypergeometric2F1[1/3, 2/3, 5/3, ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2])] + (1 - ((1 + I) + (1 - I)*Tan[(e + f*x)/2])/(2 + 2*Tan[(e + f*x)/2]))^(-1/3)))/((1 + I) + (1 - I)*Tan[(e + f*x)/2])))/(4*(1 + Tan[(e + f*x)/2]))))","C",0
119,1,128,551,0.7884955,"\int \frac{\tan ^4(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]","\frac{973 \sqrt{2} \cos (e+f x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)+\sqrt{1-\sin (e+f x)} \sec ^3(e+f x) (22 \sin (e+f x)-128 \sin (3 (e+f x))-64 \cos (2 (e+f x))-49)}{495 f \sqrt{1-\sin (e+f x)} \sqrt[3]{a (\sin (e+f x)+1)}}","\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{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{\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*Sqrt[2]*Cos[e + f*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*e + Pi + 2*f*x)/4]^2] + Sec[e + f*x]^3*Sqrt[1 - Sin[e + f*x]]*(-49 - 64*Cos[2*(e + f*x)] + 22*Sin[e + f*x] - 128*Sin[3*(e + f*x)]))/(495*f*Sqrt[1 - Sin[e + f*x]]*(a*(1 + Sin[e + f*x]))^(1/3))","C",1
120,1,100,126,0.4993876,"\int \frac{\tan ^2(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]","\frac{\sqrt{2-2 \sin (e+f x)} (4 \tan (e+f x)+\sec (e+f x))-22 \cos (e+f x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)}{5 f \sqrt{2-2 \sin (e+f x)} \sqrt[3]{a (\sin (e+f x)+1)}}","\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,"(-22*Cos[e + f*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*e + Pi + 2*f*x)/4]^2] + Sqrt[2 - 2*Sin[e + f*x]]*(Sec[e + f*x] + 4*Tan[e + f*x]))/(5*f*Sqrt[2 - 2*Sin[e + f*x]]*(a*(1 + Sin[e + f*x]))^(1/3))","A",1
121,0,0,80,12.570966,"\int \frac{\cot ^2(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]","\int \frac{\cot ^2(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","\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,"Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3), x]","F",-1
122,0,0,80,8.9419627,"\int \frac{\cot ^4(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]","\int \frac{\cot ^4(e+f x)}{\sqrt[3]{a+a \sin (e+f x)}} \, dx","\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,"Integrate[Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3), x]","F",-1
123,1,5199,269,58.5041344,"\int (a+a \sin (e+f x))^3 (g \tan (e+f x))^p \, dx","Integrate[(a + a*Sin[e + f*x])^3*(g*Tan[e + f*x])^p,x]","\text{Result too large to show}","\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{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{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)}",1,"Result too large to show","C",0
124,1,2054,187,18.1421186,"\int (a+a \sin (e+f x))^2 (g \tan (e+f x))^p \, dx","Integrate[(a + a*Sin[e + f*x])^2*(g*Tan[e + f*x])^p,x]","\text{Result too large to show}","\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,"(2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*(a + a*Sin[e + f*x])^2*Tan[(e + f*x)/2]*((2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])*(g*Tan[e + f*x])^p*(Cos[(e + f*x)/2]^4*Tan[e + f*x]^p + 4*Cos[(e + f*x)/2]^3*Sin[(e + f*x)/2]*Tan[e + f*x]^p + 6*Cos[(e + f*x)/2]^2*Sin[(e + f*x)/2]^2*Tan[e + f*x]^p + 4*Cos[(e + f*x)/2]*Sin[(e + f*x)/2]^3*Tan[e + f*x]^p + Sin[(e + f*x)/2]^4*Tan[e + f*x]^p))/(f*(1 + p)*(2 + p)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Sec[e + f*x]^2*Tan[(e + f*x)/2]*((2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])*Tan[e + f*x]^(-1 + p))/((1 + p)*(2 + p)) + (Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*((2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + p)*Tan[(e + f*x)/2]*((2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Tan[(e + f*x)/2]*(2*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2 + 4*(1 + p)*Tan[(e + f*x)/2]*((-2*(1 + p/2)*AppellF1[2 + p/2, p, 3, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) + ((1 + p/2)*p*AppellF1[2 + p/2, 1 + p, 2, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2)) + (2 + p)*(-(((1 + p)*AppellF1[1 + (1 + p)/2, p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 1, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + 4*(2 + p)*((-2*(1 + p)*AppellF1[1 + (1 + p)/2, p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) - 4*(2 + p)*((-3*(1 + p)*AppellF1[1 + (1 + p)/2, p, 4, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)))*Tan[e + f*x]^p)/((1 + p)*(2 + p))))","C",0
125,0,0,129,2.0577059,"\int (a+a \sin (e+f x)) (g \tan (e+f x))^p \, dx","Integrate[(a + a*Sin[e + f*x])*(g*Tan[e + f*x])^p,x]","\int (a+a \sin (e+f x)) (g \tan (e+f x))^p \, dx","\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,"Integrate[(a + a*Sin[e + f*x])*(g*Tan[e + f*x])^p, x]","F",-1
126,1,232,108,3.9248131,"\int \frac{(g \tan (e+f x))^p}{a+a \sin (e+f x)} \, dx","Integrate[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x]),x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 (g \tan (e+f x))^p \left(\left(p^2+5 p+6\right) \, _2F_1\left(\frac{p+1}{2},p+2;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-(p+1) \tan \left(\frac{1}{2} (e+f x)\right) \left(2 (p+3) \, _2F_1\left(\frac{p+2}{2},p+2;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-(p+2) \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(p+2,\frac{p+3}{2};\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right) \left(\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)^p}{f (p+1) (p+2) (p+3) (a \sin (e+f x)+a)}","\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,"(2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Tan[(e + f*x)/2]*((6 + 5*p + p^2)*Hypergeometric2F1[(1 + p)/2, 2 + p, (3 + p)/2, Tan[(e + f*x)/2]^2] - (1 + p)*Tan[(e + f*x)/2]*(2*(3 + p)*Hypergeometric2F1[(2 + p)/2, 2 + p, (4 + p)/2, Tan[(e + f*x)/2]^2] - (2 + p)*Hypergeometric2F1[2 + p, (3 + p)/2, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]))*(g*Tan[e + f*x])^p)/(f*(1 + p)*(2 + p)*(3 + p)*(a + a*Sin[e + f*x]))","B",0
127,1,667,138,13.9562941,"\int \frac{(g \tan (e+f x))^p}{(a+a \sin (e+f x))^2} \, dx","Integrate[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^2,x]","\frac{2^{p+1} \tan \left(\frac{1}{2} (e+f x)\right) \left(1-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)^p \left(-\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}\right)^p \tan ^{-p}(e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^4 (g \tan (e+f x))^p \left(\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(p+2,\frac{p+3}{2};\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+3}-\frac{6 \tan ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+3}{2},p+3;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+3}+\frac{12 \tan ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+3}{2},p+4;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+3}-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+2}{2},p+2;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+2}+\frac{6 \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+2}{2},p+3;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+2}-\frac{8 \tan \left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+2}{2},p+4;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+2}+\frac{\, _2F_1\left(\frac{p+1}{2},p+2;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}-\frac{2 \, _2F_1\left(\frac{p+1}{2},p+3;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}+\frac{2 \, _2F_1\left(\frac{p+1}{2},p+4;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}+\frac{2 \tan ^4\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(p+4,\frac{p+5}{2};\frac{p+7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+5}+\frac{2 \tan ^3\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(p+3,\frac{p+4}{2};\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+4}-\frac{8 \tan ^3\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(\frac{p+4}{2},p+4;\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+4}\right)}{f (a \sin (e+f x)+a)^2}","\frac{2 (g \tan (e+f x))^{p+3}}{a^2 f g^3 (p+3)}-\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{(g \tan (e+f x))^{p+1}}{a^2 f g (p+1)}",1,"(2^(1 + p)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Tan[(e + f*x)/2]*(1 - Tan[(e + f*x)/2]^2)^p*(-(Tan[(e + f*x)/2]/(-1 + Tan[(e + f*x)/2]^2)))^p*(Hypergeometric2F1[(1 + p)/2, 2 + p, (3 + p)/2, Tan[(e + f*x)/2]^2]/(1 + p) - (2*Hypergeometric2F1[(1 + p)/2, 3 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) + (2*Hypergeometric2F1[(1 + p)/2, 4 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) - (2*Hypergeometric2F1[(2 + p)/2, 2 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) + (6*Hypergeometric2F1[(2 + p)/2, 3 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) - (8*Hypergeometric2F1[(2 + p)/2, 4 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) + (Hypergeometric2F1[2 + p, (3 + p)/2, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) - (6*Hypergeometric2F1[(3 + p)/2, 3 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) + (12*Hypergeometric2F1[(3 + p)/2, 4 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) + (2*Hypergeometric2F1[3 + p, (4 + p)/2, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) - (8*Hypergeometric2F1[(4 + p)/2, 4 + p, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) + (2*Hypergeometric2F1[4 + p, (5 + p)/2, (7 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(5 + p))*(g*Tan[e + f*x])^p)/(f*(a + a*Sin[e + f*x])^2*Tan[e + f*x]^p)","B",0
128,1,1276,248,27.8155153,"\int \frac{(g \tan (e+f x))^p}{(a+a \sin (e+f x))^3} \, dx","Integrate[(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^3,x]","\frac{2^{p+1} \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)^6 \tan \left(\frac{1}{2} (e+f x)\right) \left(1-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)^p \left(-\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)-1}\right)^p \left(\frac{4 \, _2F_1\left(p+6,\frac{p+7}{2};\frac{p+9}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^6\left(\frac{1}{2} (e+f x)\right)}{p+7}+\frac{8 \, _2F_1\left(\frac{p}{2}+3,p+5;\frac{p}{2}+4;\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^5\left(\frac{1}{2} (e+f x)\right)}{p+6}-\frac{24 \, _2F_1\left(\frac{p+6}{2},p+6;\frac{p+8}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^5\left(\frac{1}{2} (e+f x)\right)}{p+6}+\frac{8 \, _2F_1\left(p+4,\frac{p+5}{2};\frac{p+7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^4\left(\frac{1}{2} (e+f x)\right)}{p+5}-\frac{40 \, _2F_1\left(\frac{p+5}{2},p+5;\frac{p+7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^4\left(\frac{1}{2} (e+f x)\right)}{p+5}+\frac{60 \, _2F_1\left(\frac{p+5}{2},p+6;\frac{p+7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^4\left(\frac{1}{2} (e+f x)\right)}{p+5}+\frac{4 \, _2F_1\left(p+3,\frac{p+4}{2};\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^3\left(\frac{1}{2} (e+f x)\right)}{p+4}-\frac{32 \, _2F_1\left(\frac{p+4}{2},p+4;\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^3\left(\frac{1}{2} (e+f x)\right)}{p+4}+\frac{80 \, _2F_1\left(\frac{p+4}{2},p+5;\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^3\left(\frac{1}{2} (e+f x)\right)}{p+4}-\frac{80 \, _2F_1\left(\frac{p+4}{2},p+6;\frac{p+6}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^3\left(\frac{1}{2} (e+f x)\right)}{p+4}+\frac{\, _2F_1\left(p+2,\frac{p+3}{2};\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+3}-\frac{12 \, _2F_1\left(\frac{p+3}{2},p+3;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+3}+\frac{48 \, _2F_1\left(\frac{p+3}{2},p+4;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+3}-\frac{80 \, _2F_1\left(\frac{p+3}{2},p+5;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+3}+\frac{60 \, _2F_1\left(\frac{p+3}{2},p+6;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+3}-\frac{2 \, _2F_1\left(\frac{p+2}{2},p+2;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+2}+\frac{12 \, _2F_1\left(\frac{p+2}{2},p+3;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+2}-\frac{32 \, _2F_1\left(\frac{p+2}{2},p+4;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+2}+\frac{40 \, _2F_1\left(\frac{p+2}{2},p+5;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+2}-\frac{24 \, _2F_1\left(\frac{p+2}{2},p+6;\frac{p+4}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+2}+\frac{\, _2F_1\left(\frac{p+1}{2},p+2;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}-\frac{4 \, _2F_1\left(\frac{p+1}{2},p+3;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}+\frac{8 \, _2F_1\left(\frac{p+1}{2},p+4;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}-\frac{8 \, _2F_1\left(\frac{p+1}{2},p+5;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}+\frac{4 \, _2F_1\left(\frac{p+1}{2},p+6;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{p+1}\right) \tan ^{-p}(e+f x) (g \tan (e+f x))^p}{f (\sin (e+f x) a+a)^3}","\frac{4 (g \tan (e+f x))^{p+5}}{a^3 f g^5 (p+5)}-\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{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{(g \tan (e+f x))^{p+1}}{a^3 f g (p+1)}",1,"(2^(1 + p)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Tan[(e + f*x)/2]*(1 - Tan[(e + f*x)/2]^2)^p*(-(Tan[(e + f*x)/2]/(-1 + Tan[(e + f*x)/2]^2)))^p*(Hypergeometric2F1[(1 + p)/2, 2 + p, (3 + p)/2, Tan[(e + f*x)/2]^2]/(1 + p) - (4*Hypergeometric2F1[(1 + p)/2, 3 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) + (8*Hypergeometric2F1[(1 + p)/2, 4 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) - (8*Hypergeometric2F1[(1 + p)/2, 5 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) + (4*Hypergeometric2F1[(1 + p)/2, 6 + p, (3 + p)/2, Tan[(e + f*x)/2]^2])/(1 + p) - (2*Hypergeometric2F1[(2 + p)/2, 2 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) + (12*Hypergeometric2F1[(2 + p)/2, 3 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) - (32*Hypergeometric2F1[(2 + p)/2, 4 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) + (40*Hypergeometric2F1[(2 + p)/2, 5 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) - (24*Hypergeometric2F1[(2 + p)/2, 6 + p, (4 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(2 + p) + (Hypergeometric2F1[2 + p, (3 + p)/2, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) - (12*Hypergeometric2F1[(3 + p)/2, 3 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) + (48*Hypergeometric2F1[(3 + p)/2, 4 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) - (80*Hypergeometric2F1[(3 + p)/2, 5 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) + (60*Hypergeometric2F1[(3 + p)/2, 6 + p, (5 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(3 + p) + (4*Hypergeometric2F1[3 + p, (4 + p)/2, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) - (32*Hypergeometric2F1[(4 + p)/2, 4 + p, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) + (80*Hypergeometric2F1[(4 + p)/2, 5 + p, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) - (80*Hypergeometric2F1[(4 + p)/2, 6 + p, (6 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(4 + p) + (8*Hypergeometric2F1[4 + p, (5 + p)/2, (7 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(5 + p) - (40*Hypergeometric2F1[(5 + p)/2, 5 + p, (7 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(5 + p) + (60*Hypergeometric2F1[(5 + p)/2, 6 + p, (7 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(5 + p) + (8*Hypergeometric2F1[3 + p/2, 5 + p, 4 + p/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^5)/(6 + p) - (24*Hypergeometric2F1[(6 + p)/2, 6 + p, (8 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^5)/(6 + p) + (4*Hypergeometric2F1[6 + p, (7 + p)/2, (9 + p)/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^6)/(7 + p))*(g*Tan[e + f*x])^p)/(f*(a + a*Sin[e + f*x])^3*Tan[e + f*x]^p)","B",0
129,1,367,111,2.178053,"\int (a+a \sin (e+f x))^m (g \tan (e+f x))^p \, dx","Integrate[(a + a*Sin[e + f*x])^m*(g*Tan[e + f*x])^p,x]","-\frac{2 (p-3) \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \cos ^3\left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (g \tan (e+f x))^p F_1\left(\frac{1-p}{2};-p,m+1;\frac{3-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (p-1) \left((p-3) \cos ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) F_1\left(\frac{1-p}{2};-p,m+1;\frac{3-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+2 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(p F_1\left(\frac{3-p}{2};1-p,m+1;\frac{5-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+1) F_1\left(\frac{3-p}{2};-p,m+2;\frac{5-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)\right)}","\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,"(-2*(-3 + p)*AppellF1[(1 - p)/2, -p, 1 + m, (3 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[(2*e - Pi + 2*f*x)/4]^3*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e - Pi + 2*f*x)/4]*(g*Tan[e + f*x])^p)/(f*(-1 + p)*((-3 + p)*AppellF1[(1 - p)/2, -p, 1 + m, (3 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[(2*e - Pi + 2*f*x)/4]^2 + 2*(p*AppellF1[(3 - p)/2, 1 - p, 1 + m, (5 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m)*AppellF1[(3 - p)/2, -p, 2 + m, (5 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Sin[(2*e - Pi + 2*f*x)/4]^2))","B",0
130,1,105,163,0.2571126,"\int (a+a \sin (e+f x))^m \tan ^3(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^3,x]","\frac{a (a (\sin (e+f x)+1))^{m-1} \left(-m (m+4) (\sin (e+f x)-1) \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)+4 (m-1) \sin ^2(e+f x)+4 m \sin (e+f x)-2 \left(m^2+3 m-2\right)\right)}{4 f (m-1) m (\sin (e+f x)-1)}","-\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*(a*(1 + Sin[e + f*x]))^(-1 + m)*(-2*(-2 + 3*m + m^2) - m*(4 + m)*Hypergeometric2F1[1, -1 + m, m, (1 + Sin[e + f*x])/2]*(-1 + Sin[e + f*x]) + 4*m*Sin[e + f*x] + 4*(-1 + m)*Sin[e + f*x]^2))/(4*f*(-1 + m)*m*(-1 + Sin[e + f*x]))","A",1
131,1,63,72,0.0693979,"\int (a+a \sin (e+f x))^m \tan (e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x],x]","\frac{(a (\sin (e+f x)+1))^m \left(m (\sin (e+f x)+1) \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)-2 (m+1)\right)}{4 f m (m+1)}","\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*(1 + Sin[e + f*x]))^m*(-2*(1 + m) + m*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[e + f*x])/2]*(1 + Sin[e + f*x])))/(4*f*m*(1 + m))","A",1
132,1,43,43,0.0549198,"\int \cot (e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[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",1
133,1,68,83,0.1923586,"\int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^3*(a + a*Sin[e + f*x])^m,x]","-\frac{(\sin (e+f x)+1)^2 (a (\sin (e+f x)+1))^m \left((m+2) \csc ^2(e+f x)-(m-2) \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)\right)}{2 f (m+2)}","-\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,"-1/2*(((2 + m)*Csc[e + f*x]^2 - (-2 + m)*Hypergeometric2F1[2, 2 + m, 3 + m, 1 + Sin[e + f*x]])*(1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^m)/(f*(2 + m))","A",1
134,1,83,123,0.2633074,"\int \cot ^5(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^5*(a + a*Sin[e + f*x])^m,x]","-\frac{(\sin (e+f x)+1)^3 (a (\sin (e+f x)+1))^m \left(\left(m^2-9 m+12\right) \, _2F_1(3,m+3;m+4;\sin (e+f x)+1)+(m+3) \csc ^3(e+f x) (3 \csc (e+f x)+m-9)\right)}{12 f (m+3)}","-\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,"-1/12*(((3 + m)*Csc[e + f*x]^3*(-9 + m + 3*Csc[e + f*x]) + (12 - 9*m + m^2)*Hypergeometric2F1[3, 3 + m, 4 + m, 1 + Sin[e + f*x]])*(1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^m)/(f*(3 + m))","A",1
135,0,0,311,1.1229909,"\int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]","\int (a+a \sin (e+f x))^m \tan ^4(e+f x) \, dx","-\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,"Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^4, x]","F",-1
136,1,4043,157,6.32458,"\int (a+a \sin (e+f x))^m \tan ^2(e+f x) \, dx","Integrate[(a + a*Sin[e + f*x])^m*Tan[e + f*x]^2,x]","\text{Result too large to show}","\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,"(-2*Cos[(-e + Pi/2 - f*x)/2]*Hypergeometric2F1[1/2, (1 + 2*m)/2, (3 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2]) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(a + a*Sin[e + f*x])^m*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/(4*f*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^2*(-1/2*(m*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2))) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Csc[(-e + Pi/2 - f*x)/4]^2*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/8 + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(-(m*AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]) - (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(m*AppellF1[1/2, 1 - 2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + m*AppellF1[1/2, -2*m, 1 + 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)) + (3*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*m*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-2*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + 3*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3) - 4*m*Tan[(-e + Pi/2 - f*x)/4]^2*((-6*m*AppellF1[5/2, 1 - 2*m, 1 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/5 + (3*(1 - 2*m)*AppellF1[5/2, 2 - 2*m, 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10 - (3*(1 + 2*m)*AppellF1[5/2, -2*m, 2 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10)))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)^2))/2)) + (Hypergeometric2F1[1/2, (-1 + 2*m)/2, (1 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*(a + a*Sin[e + f*x])^m*Tan[(-e + Pi/2 - f*x)/2])/(2*f*(-1 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2])","C",0
137,1,90,74,0.1543709,"\int (a+a \sin (e+f x))^m \, dx","Integrate[(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{2} \cos (e+f x) (a (\sin (e+f x)+1))^m \, _2F_1\left(\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{4} \cos ^2(e+f x) \csc ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{(2 f m+f) \sqrt{1-\sin (e+f 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}",1,"(Sqrt[2]*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, (Cos[e + f*x]^2*Csc[(2*e - Pi + 2*f*x)/4]^2)/4]*(a*(1 + Sin[e + f*x]))^m)/((f + 2*f*m)*Sqrt[1 - Sin[e + f*x]])","A",1
138,1,5048,89,26.4468243,"\int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^m,x]","\text{Result too large to show}","\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,"Result too large to show","C",0
139,0,0,89,0.7810548,"\int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^m,x]","\int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx","\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,"Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^m, x]","F",-1
140,1,77,88,0.1262302,"\int (a+b \sin (c+d x)) \tan ^3(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}-\frac{b \sin (c+d x) \tan ^2(c+d x)}{d}-\frac{3 b \left(\tanh ^{-1}(\sin (c+d x))-\tan (c+d x) \sec (c+d x)\right)}{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,"-((b*Sin[c + d*x]*Tan[c + d*x]^2)/d) - (3*b*(ArcTanh[Sin[c + d*x]] - Sec[c + d*x]*Tan[c + d*x]))/(2*d) + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
141,1,38,55,0.0174714,"\int (a+b \sin (c+d x)) \tan (c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x],x]","-\frac{a \log (\cos (c+d x))}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\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,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d - (b*Sin[c + d*x])/d","A",1
142,1,43,24,0.0357217,"\int \cot (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{b \sin (c) \cos (d x)}{d}+\frac{b \cos (c) \sin (d x)}{d}","\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}",1,"(a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d + (b*Cos[d*x]*Sin[c])/d + (b*Cos[c]*Sin[d*x])/d","A",1
143,1,60,54,0.2118759,"\int \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 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*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) - (b*Sin[c + d*x])/d","A",1
144,1,87,81,0.2436637,"\int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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 - (b*Csc[c + d*x]^3)/(3*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d) + (b*Sin[c + d*x])/d","A",1
145,1,81,72,0.0374632,"\int (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}-\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*ArcTan[Tan[c + d*x]])/d - (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",1
146,1,47,38,0.0350725,"\int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\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*ArcTan[Tan[c + d*x]])/d) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
147,1,75,41,0.0372564,"\int \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{b \cos (c+d x)}{d}+\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"(b*Cos[c + d*x])/d - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (b*Log[Cos[(c + d*x)/2]])/d + (b*Log[Sin[(c + d*x)/2]])/d","C",1
148,1,125,82,0.0456284,"\int \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{b \cos (c+d x)}{d}-\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-((b*Cos[c + d*x])/d) - (b*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (3*b*Log[Cos[(c + d*x)/2]])/(2*d) - (3*b*Log[Sin[(c + d*x)/2]])/(2*d) + (b*Sec[(c + d*x)/2]^2)/(8*d)","C",1
149,1,164,122,0.0540227,"\int \cot ^6(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}+\frac{b \cos (c+d x)}{d}-\frac{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{9 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{9 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{15 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{15 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"(b*Cos[c + d*x])/d + (9*b*Csc[(c + d*x)/2]^2)/(32*d) - (b*Csc[(c + d*x)/2]^4)/(64*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) - (15*b*Log[Cos[(c + d*x)/2]])/(8*d) + (15*b*Log[Sin[(c + d*x)/2]])/(8*d) - (9*b*Sec[(c + d*x)/2]^2)/(32*d) + (b*Sec[(c + d*x)/2]^4)/(64*d)","C",1
150,1,108,111,0.4413623,"\int (a+b \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{\frac{(a-b)^2}{\sin (c+d x)+1}+8 a b \sin (c+d x)-\frac{(a+b)^2}{\sin (c+d x)-1}+2 (a-2 b) (a-b) \log (\sin (c+d x)+1)+2 (a+b) (a+2 b) \log (1-\sin (c+d x))+2 b^2 \sin ^2(c+d x)}{4 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,"(2*(a + b)*(a + 2*b)*Log[1 - Sin[c + d*x]] + 2*(a - 2*b)*(a - b)*Log[1 + Sin[c + d*x]] - (a + b)^2/(-1 + Sin[c + d*x]) + 8*a*b*Sin[c + d*x] + 2*b^2*Sin[c + d*x]^2 + (a - b)^2/(1 + Sin[c + d*x]))/(4*d)","A",1
151,1,64,78,0.1315365,"\int (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{4 a b \sin (c+d x)+(a-b)^2 \log (\sin (c+d x)+1)+(a+b)^2 \log (1-\sin (c+d x))+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,"-1/2*((a + b)^2*Log[1 - Sin[c + d*x]] + (a - b)^2*Log[1 + Sin[c + d*x]] + 4*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2)/d","A",1
152,1,46,46,0.0245419,"\int \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[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",1
153,1,70,84,0.2589071,"\int \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-b^2\right) \log (\sin (c+d x))+a^2 \csc ^2(c+d x)+4 a b \sin (c+d x)+4 a b \csc (c+d x)+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,"-1/2*(4*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - b^2)*Log[Sin[c + d*x]] + 4*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2)/d","A",1
154,1,107,126,0.7323083,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{6 \left(2 a^2-b^2\right) \csc ^2(c+d x)+6 \left(2 \left(a^2-2 b^2\right) \log (\sin (c+d x))+4 a b \sin (c+d x)+b^2 \sin ^2(c+d x)\right)-3 a^2 \csc ^4(c+d x)-8 a b \csc ^3(c+d x)+48 a b \csc (c+d x)}{12 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,"(48*a*b*Csc[c + d*x] + 6*(2*a^2 - b^2)*Csc[c + d*x]^2 - 8*a*b*Csc[c + d*x]^3 - 3*a^2*Csc[c + d*x]^4 + 6*(2*(a^2 - 2*b^2)*Log[Sin[c + d*x]] + 4*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2))/(12*d)","A",1
155,1,176,149,0.723935,"\int (a+b \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{\sec ^3(c+d x) \left(-36 \left(2 a^2+5 b^2\right) (c+d x) \cos (c+d x)+32 a^2 \sin (3 (c+d x))-24 a^2 c \cos (3 (c+d x))-24 a^2 d x \cos (3 (c+d x))+288 a b \cos (2 (c+d x))+24 a b \cos (4 (c+d x))+200 a b+30 b^2 \sin (c+d x)+65 b^2 \sin (3 (c+d x))+3 b^2 \sin (5 (c+d x))-60 b^2 c \cos (3 (c+d x))-60 b^2 d x \cos (3 (c+d x))\right)}{96 d}","\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,"-1/96*(Sec[c + d*x]^3*(200*a*b - 36*(2*a^2 + 5*b^2)*(c + d*x)*Cos[c + d*x] + 288*a*b*Cos[2*(c + d*x)] - 24*a^2*c*Cos[3*(c + d*x)] - 60*b^2*c*Cos[3*(c + d*x)] - 24*a^2*d*x*Cos[3*(c + d*x)] - 60*b^2*d*x*Cos[3*(c + d*x)] + 24*a*b*Cos[4*(c + d*x)] + 30*b^2*Sin[c + d*x] + 32*a^2*Sin[3*(c + d*x)] + 65*b^2*Sin[3*(c + d*x)] + 3*b^2*Sin[5*(c + d*x)]))/d","A",1
156,1,77,94,0.4848932,"\int (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{-4 \left(2 a^2+3 b^2\right) (c+d x)+\left(8 a^2+9 b^2\right) \tan (c+d x)+b \sec (c+d x) (8 a \cos (2 (c+d x))+24 a+b \sin (3 (c+d x)))}{8 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 (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,"(-4*(2*a^2 + 3*b^2)*(c + d*x) + b*Sec[c + d*x]*(24*a + 8*a*Cos[2*(c + d*x)] + b*Sin[3*(c + d*x)]) + (8*a^2 + 9*b^2)*Tan[c + d*x])/(8*d)","A",1
157,1,116,78,0.41108,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 \cot \left(\frac{1}{2} (c+d x)\right)-4 a^2 c-4 a^2 d x+8 a b \cos (c+d x)+8 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 \sin (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 d}","-\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,"(-4*a^2*c + 2*b^2*c - 4*a^2*d*x + 2*b^2*d*x + 8*a*b*Cos[c + d*x] - 2*a^2*Cot[(c + d*x)/2] - 8*a*b*Log[Cos[(c + d*x)/2]] + 8*a*b*Log[Sin[(c + d*x)/2]] + b^2*Sin[2*(c + d*x)] + 2*a^2*Tan[(c + d*x)/2])/(4*d)","A",1
158,1,293,133,6.2142732,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-3 b^2\right) (c+d x)}{2 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}-\frac{a^2 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}-\frac{2 a b \cos (c+d x)}{d}-\frac{a b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}+\frac{a b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}-\frac{3 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b^2 \sin (2 (c+d x))}{4 d}","-\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,"((2*a^2 - 3*b^2)*(c + d*x))/(2*d) - (2*a*b*Cos[c + d*x])/d + ((4*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*d) - (a*b*Csc[(c + d*x)/2]^2)/(4*d) - (a^2*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + (3*a*b*Log[Cos[(c + d*x)/2]])/d - (3*a*b*Log[Sin[(c + d*x)/2]])/d + (a*b*Sec[(c + d*x)/2]^2)/(4*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*d) - (b^2*Sin[2*(c + d*x)])/(4*d) + (a^2*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)","B",0
159,1,351,202,1.1204278,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{\left(560 b^2-368 a^2\right) \cot \left(\frac{1}{2} (c+d x)\right)+368 a^2 \tan \left(\frac{1}{2} (c+d x)\right)-\frac{3}{2} a^2 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+96 a^2 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+\frac{41}{2} a^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-328 a^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-480 a^2 c-480 a^2 d x+960 a b \cos (c+d x)-15 a b \csc ^4\left(\frac{1}{2} (c+d x)\right)+270 a b \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 a b \sec ^4\left(\frac{1}{2} (c+d x)\right)-270 a b \sec ^2\left(\frac{1}{2} (c+d x)\right)+1800 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1800 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 b^2 \sin (2 (c+d x))-560 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-10 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+160 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+1200 b^2 c+1200 b^2 d x}{480 d}","-\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,"(-480*a^2*c + 1200*b^2*c - 480*a^2*d*x + 1200*b^2*d*x + 960*a*b*Cos[c + d*x] + (-368*a^2 + 560*b^2)*Cot[(c + d*x)/2] + 270*a*b*Csc[(c + d*x)/2]^2 - 15*a*b*Csc[(c + d*x)/2]^4 - 1800*a*b*Log[Cos[(c + d*x)/2]] + 1800*a*b*Log[Sin[(c + d*x)/2]] - 270*a*b*Sec[(c + d*x)/2]^2 + 15*a*b*Sec[(c + d*x)/2]^4 - 328*a^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 160*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*a^2*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + (41*a^2*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 10*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - (3*a^2*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 120*b^2*Sin[2*(c + d*x)] + 368*a^2*Tan[(c + d*x)/2] - 560*b^2*Tan[(c + d*x)/2])/(480*d)","A",1
160,1,141,150,0.2486524,"\int (a+b \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{12 b \left(3 a^2+2 b^2\right) \sin (c+d x)+18 a b^2 \sin ^2(c+d x)+\frac{3 (a-b)^3}{\sin (c+d x)+1}-\frac{3 (a+b)^3}{\sin (c+d x)-1}+3 (2 a-5 b) (a-b)^2 \log (\sin (c+d x)+1)+3 (a+b)^2 (2 a+5 b) \log (1-\sin (c+d x))+4 b^3 \sin ^3(c+d x)}{12 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,"(3*(a + b)^2*(2*a + 5*b)*Log[1 - Sin[c + d*x]] + 3*(2*a - 5*b)*(a - b)^2*Log[1 + Sin[c + d*x]] - (3*(a + b)^3)/(-1 + Sin[c + d*x]) + 12*b*(3*a^2 + 2*b^2)*Sin[c + d*x] + 18*a*b^2*Sin[c + d*x]^2 + 4*b^3*Sin[c + d*x]^3 + (3*(a - b)^3)/(1 + Sin[c + d*x]))/(12*d)","A",1
161,1,90,105,0.199334,"\int (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{6 b \left(3 a^2+b^2\right) \sin (c+d x)+9 a b^2 \sin ^2(c+d x)+3 \left((a-b)^3 \log (\sin (c+d x)+1)+(a+b)^3 \log (1-\sin (c+d x))\right)+2 b^3 \sin ^3(c+d x)}{6 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,"-1/6*(3*((a + b)^3*Log[1 - Sin[c + d*x]] + (a - b)^3*Log[1 + Sin[c + d*x]]) + 6*b*(3*a^2 + b^2)*Sin[c + d*x] + 9*a*b^2*Sin[c + d*x]^2 + 2*b^3*Sin[c + d*x]^3)/d","A",1
162,1,67,67,0.0269456,"\int \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^2 b \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{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^2 b \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",1
163,1,97,116,0.2889955,"\int \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{3 a^3 \csc ^2(c+d x)-6 b \left(b^2-3 a^2\right) \sin (c+d x)+6 a \left(a^2-3 b^2\right) \log (\sin (c+d x))+18 a^2 b \csc (c+d x)+9 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)}{6 d}","-\frac{a^3 \csc ^2(c+d x)}{2 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{3 a b^2 \sin ^2(c+d x)}{2 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"-1/6*(18*a^2*b*Csc[c + d*x] + 3*a^3*Csc[c + d*x]^2 + 6*a*(a^2 - 3*b^2)*Log[Sin[c + d*x]] - 6*b*(-3*a^2 + b^2)*Sin[c + d*x] + 9*a*b^2*Sin[c + d*x]^2 + 2*b^3*Sin[c + d*x]^3)/d","A",1
164,1,144,165,1.0372638,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{-3 a^3 \csc ^4(c+d x)+6 a \left(2 a^2-3 b^2\right) \csc ^2(c+d x)-12 b \left(b^2-6 a^2\right) \csc (c+d x)+2 \left(6 b \left(3 a^2-2 b^2\right) \sin (c+d x)+6 a \left(a^2-6 b^2\right) \log (\sin (c+d x))+9 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)\right)-12 a^2 b \csc ^3(c+d x)}{12 d}","-\frac{a^3 \csc ^4(c+d x)}{4 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{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(-12*b*(-6*a^2 + b^2)*Csc[c + d*x] + 6*a*(2*a^2 - 3*b^2)*Csc[c + d*x]^2 - 12*a^2*b*Csc[c + d*x]^3 - 3*a^3*Csc[c + d*x]^4 + 2*(6*a*(a^2 - 6*b^2)*Log[Sin[c + d*x]] + 6*b*(3*a^2 - 2*b^2)*Sin[c + d*x] + 9*a*b^2*Sin[c + d*x]^2 + 2*b^3*Sin[c + d*x]^3))/(12*d)","A",1
165,1,226,220,0.7075627,"\int (a+b \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{\sec ^3(c+d x) \left(-32 a^3 \sin (3 (c+d x))+24 a^3 c \cos (3 (c+d x))+24 a^3 d x \cos (3 (c+d x))-3 \left(144 a^2 b+91 b^3\right) \cos (2 (c+d x))+36 a \left(2 a^2+15 b^2\right) (c+d x) \cos (c+d x)-36 a^2 b \cos (4 (c+d x))-300 a^2 b-90 a b^2 \sin (c+d x)-195 a b^2 \sin (3 (c+d x))-9 a b^2 \sin (5 (c+d x))+180 a b^2 c \cos (3 (c+d x))+180 a b^2 d x \cos (3 (c+d x))-30 b^3 \cos (4 (c+d x))+b^3 \cos (6 (c+d x))-210 b^3\right)}{96 d}","\frac{a^3 \tan ^3(c+d x)}{3 d}-\frac{a^3 \tan (c+d x)}{d}+a^3 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{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,"(Sec[c + d*x]^3*(-300*a^2*b - 210*b^3 + 36*a*(2*a^2 + 15*b^2)*(c + d*x)*Cos[c + d*x] - 3*(144*a^2*b + 91*b^3)*Cos[2*(c + d*x)] + 24*a^3*c*Cos[3*(c + d*x)] + 180*a*b^2*c*Cos[3*(c + d*x)] + 24*a^3*d*x*Cos[3*(c + d*x)] + 180*a*b^2*d*x*Cos[3*(c + d*x)] - 36*a^2*b*Cos[4*(c + d*x)] - 30*b^3*Cos[4*(c + d*x)] + b^3*Cos[6*(c + d*x)] - 90*a*b^2*Sin[c + d*x] - 32*a^3*Sin[3*(c + d*x)] - 195*a*b^2*Sin[3*(c + d*x)] - 9*a*b^2*Sin[5*(c + d*x)]))/(96*d)","A",1
166,1,113,146,0.7687308,"\int (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{3 a \left(\left(8 a^2+27 b^2\right) \tan (c+d x)-4 \left(2 a^2+9 b^2\right) (c+d x)\right)+b \sec (c+d x) \left(4 \left(9 a^2+5 b^2\right) \cos (2 (c+d x))+108 a^2+9 a b \sin (3 (c+d x))-b^2 \cos (4 (c+d x))+45 b^2\right)}{24 d}","\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\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,"(b*Sec[c + d*x]*(108*a^2 + 45*b^2 + 4*(9*a^2 + 5*b^2)*Cos[2*(c + d*x)] - b^2*Cos[4*(c + d*x)] + 9*a*b*Sin[3*(c + d*x)]) + 3*a*(-4*(2*a^2 + 9*b^2)*(c + d*x) + (8*a^2 + 27*b^2)*Tan[c + d*x]))/(24*d)","A",1
167,1,143,102,1.3226833,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-6 a^3 \cot \left(\frac{1}{2} (c+d x)\right)+\left(36 a^2 b-3 b^3\right) \cos (c+d x)+6 a \left(a^2 \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 c-2 a^2 d x+6 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b^2 c+3 b^2 d x\right)+9 a b^2 \sin (2 (c+d x))-b^3 \cos (3 (c+d x))}{12 d}","-\frac{a^3 \cot (c+d x)}{d}+a^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{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,"((36*a^2*b - 3*b^3)*Cos[c + d*x] - b^3*Cos[3*(c + d*x)] - 6*a^3*Cot[(c + d*x)/2] + 9*a*b^2*Sin[2*(c + d*x)] + 6*a*(-2*a^2*c + 3*b^2*c - 2*a^2*d*x + 3*b^2*d*x - 6*a*b*Log[Cos[(c + d*x)/2]] + 6*a*b*Log[Sin[(c + d*x)/2]] + a^2*Tan[(c + d*x)/2]))/(12*d)","A",1
168,1,355,194,6.2336629,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^3 \cos \left(\frac{1}{2} (c+d x)\right)-9 a b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 a b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}-\frac{a^3 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{a^3 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{\left(2 b^3-9 a^2 b\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{\left(9 a^2 b-2 b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \left(2 a^2-9 b^2\right) (c+d x)}{2 d}+\frac{b \left(5 b^2-12 a^2\right) \cos (c+d x)}{4 d}-\frac{3 a^2 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a^2 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a b^2 \sin (2 (c+d x))}{4 d}+\frac{b^3 \cos (3 (c+d x))}{12 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^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{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*(2*a^2 - 9*b^2)*(c + d*x))/(2*d) + (b*(-12*a^2 + 5*b^2)*Cos[c + d*x])/(4*d) + (b^3*Cos[3*(c + d*x)])/(12*d) + ((4*a^3*Cos[(c + d*x)/2] - 9*a*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*d) - (3*a^2*b*Csc[(c + d*x)/2]^2)/(8*d) - (a^3*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + ((9*a^2*b - 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*d) + ((-9*a^2*b + 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*d) + (3*a^2*b*Sec[(c + d*x)/2]^2)/(8*d) + (Sec[(c + d*x)/2]*(-4*a^3*Sin[(c + d*x)/2] + 9*a*b^2*Sin[(c + d*x)/2]))/(6*d) - (3*a*b^2*Sin[2*(c + d*x)])/(4*d) + (a^3*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)","A",0
169,1,346,291,2.5740603,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]","\frac{-600 a \left(2 a^2-15 b^2\right) (c+d x) \csc ^4(c+d x)+1200 b \left(4 b^2-9 a^2\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+5 \cot (c+d x) \csc ^4(c+d x) \left(-80 a^3+12 b \left(60 a^2-29 b^2\right) \sin (c+d x)+285 a b^2\right)+\csc ^5(c+d x) \left(5 \left(40 a^3-489 a b^2\right) \cos (3 (c+d x))+\left(1065 a b^2-184 a^3\right) \cos (5 (c+d x))+5 \left(-24 a^3 c \sin (5 (c+d x))-24 a^3 d x \sin (5 (c+d x))+60 a \left(2 a^2-15 b^2\right) (c+d x) \sin (3 (c+d x))-306 a^2 b \sin (4 (c+d x))+36 a^2 b \sin (6 (c+d x))+180 a b^2 c \sin (5 (c+d x))+180 a b^2 d x \sin (5 (c+d x))-9 a b^2 \cos (7 (c+d x))+122 b^3 \sin (4 (c+d x))-22 b^3 \sin (6 (c+d x))-b^3 \sin (8 (c+d x))\right)\right)}{1920 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{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{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,"(-600*a*(2*a^2 - 15*b^2)*(c + d*x)*Csc[c + d*x]^4 + 1200*b*(-9*a^2 + 4*b^2)*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + 5*Cot[c + d*x]*Csc[c + d*x]^4*(-80*a^3 + 285*a*b^2 + 12*b*(60*a^2 - 29*b^2)*Sin[c + d*x]) + Csc[c + d*x]^5*(5*(40*a^3 - 489*a*b^2)*Cos[3*(c + d*x)] + (-184*a^3 + 1065*a*b^2)*Cos[5*(c + d*x)] + 5*(-9*a*b^2*Cos[7*(c + d*x)] + 60*a*(2*a^2 - 15*b^2)*(c + d*x)*Sin[3*(c + d*x)] - 306*a^2*b*Sin[4*(c + d*x)] + 122*b^3*Sin[4*(c + d*x)] - 24*a^3*c*Sin[5*(c + d*x)] + 180*a*b^2*c*Sin[5*(c + d*x)] - 24*a^3*d*x*Sin[5*(c + d*x)] + 180*a*b^2*d*x*Sin[5*(c + d*x)] + 36*a^2*b*Sin[6*(c + d*x)] - 22*b^3*Sin[6*(c + d*x)] - b^3*Sin[8*(c + d*x)])))/(1920*d)","A",1
170,1,184,204,1.3020462,"\int \frac{\tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{\frac{16 a^5 \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{(a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{(a-b)^3}+\frac{7 a+5 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{5 b-7 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}+\frac{1}{(a-b) (\sin (c+d x)+1)^2}}{16 d}","-\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}",1,"(-(((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(a + b)^3) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 + (16*a^5*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (7*a + 5*b)/((a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])^2) + (-7*a + 5*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
171,1,117,126,0.491723,"\int \frac{\tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{-\frac{4 a^3 \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}-\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{(a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{(a-b)^2}}{4 d}","\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\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]])/(a + b)^2 + ((2*a - b)*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*a^3*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) - 1/((a + b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])))/(4*d)","A",1
172,1,87,74,0.084681,"\int \frac{\tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{a \log (a+b \sin (c+d x))+(b-a) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-(a+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a-b) (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,"((-a + b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - (a + b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a*Log[a + b*Sin[c + d*x]])/((a - b)*(a + b)*d)","A",1
173,1,34,34,0.0186914,"\int \frac{\cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[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",1
174,1,65,84,0.1584636,"\int \frac{\cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right) (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))+a^2 \csc ^2(c+d x)-2 a b \csc (c+d x)}{2 a^3 d}","\frac{b \csc (c+d x)}{a^2 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{\csc ^2(c+d x)}{2 a d}",1,"-1/2*(-2*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - b^2)*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(a^3*d)","A",1
175,1,115,148,1.0602826,"\int \frac{\cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{-3 a^4 \csc ^4(c+d x)+4 a^3 b \csc ^3(c+d x)+6 a^2 \left(2 a^2-b^2\right) \csc ^2(c+d x)+12 a b \left(b^2-2 a^2\right) \csc (c+d x)+12 \left(a^2-b^2\right)^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))}{12 a^5 d}","\frac{b \csc ^3(c+d x)}{3 a^2 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 \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"(12*a*b*(-2*a^2 + b^2)*Csc[c + d*x] + 6*a^2*(2*a^2 - b^2)*Csc[c + d*x]^2 + 4*a^3*b*Csc[c + d*x]^3 - 3*a^4*Csc[c + d*x]^4 + 12*(a^2 - b^2)^2*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(12*a^5*d)","A",1
176,1,195,177,1.400708,"\int \frac{\tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{\frac{48 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(c+d x) \left(8 a^3 \sin (3 (c+d x))+3 b \left(11 a^2-5 b^2\right) \cos (c+d x)+12 b \left(b^2-2 a^2\right) \cos (2 (c+d x))+11 a^2 b \cos (3 (c+d x))-16 a^2 b+6 a b^2 \sin (c+d x)-2 a b^2 \sin (3 (c+d x))-5 b^3 \cos (3 (c+d x))+4 b^3\right)}{(a-b)^2 (a+b)^2}}{24 d}","\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\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^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}",1,"((48*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Sec[c + d*x]^3*(-16*a^2*b + 4*b^3 + 3*b*(11*a^2 - 5*b^2)*Cos[c + d*x] + 12*b*(-2*a^2 + b^2)*Cos[2*(c + d*x)] + 11*a^2*b*Cos[3*(c + d*x)] - 5*b^3*Cos[3*(c + d*x)] + 6*a*b^2*Sin[c + d*x] + 8*a^3*Sin[3*(c + d*x)] - 2*a*b^2*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(24*d)","A",1
177,1,152,96,0.1967642,"\int \frac{\tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2-b^2} (a \sin (c+d x)+b \cos (c+d x)-b)-2 a^2 \cos (c+d x) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d (a-b) (a+b) \sqrt{a^2-b^2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\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]]*Cos[c + d*x] + Sqrt[a^2 - b^2]*(-b + b*Cos[c + d*x] + a*Sin[c + d*x]))/((a - b)*(a + b)*Sqrt[a^2 - b^2]*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
178,1,108,80,0.2446153,"\int \frac{\cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{-4 \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 \tan \left(\frac{1}{2} (c+d x)\right)-a \cot \left(\frac{1}{2} (c+d x)\right)-2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^2 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,"(-4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - a*Cot[(c + d*x)/2] + 2*b*Log[Cos[(c + d*x)/2]] - 2*b*Log[Sin[(c + d*x)/2]] + a*Tan[(c + d*x)/2])/(2*a^2*d)","A",1
179,1,350,154,6.1210968,"\int \frac{\cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\left(3 a^2 b-2 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{\left(2 b^3-3 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}","\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 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{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(2*(a^2 - b^2)^(3/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*d) + ((4*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^3*d) + (b*Csc[(c + d*x)/2]^2)/(8*a^2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a*d) + ((-3*a^2*b + 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + ((3*a^2*b - 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) - (b*Sec[(c + d*x)/2]^2)/(8*a^2*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*a^3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a*d)","B",0
180,1,504,307,1.4141807,"\int \frac{\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]","\frac{736 a^5 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^5 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+41 a^5 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-656 a^5 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 a^5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+15 a^4 b \csc ^4\left(\frac{1}{2} (c+d x)\right)-270 a^4 b \csc ^2\left(\frac{1}{2} (c+d x)\right)-15 a^4 b \sec ^4\left(\frac{1}{2} (c+d x)\right)+270 a^4 b \sec ^2\left(\frac{1}{2} (c+d x)\right)-1800 a^4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1800 a^4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1120 a^3 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-20 a^3 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 a^3 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)-120 a^2 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+2400 a^2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2400 a^2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1920 \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)-32 \left(23 a^5-35 a^3 b^2+15 a b^4\right) \cot \left(\frac{1}{2} (c+d x)\right)+480 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)-960 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+960 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{960 a^6 d}","\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 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(8 a^4-9 a^2 b^2+4 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{b \left(15 a^4-20 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(23 a^4-35 a^2 b^2+15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{\left(15 a^4-22 a^2 b^2+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^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,"(-1920*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*Cot[(c + d*x)/2] - 270*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 + 1800*a^4*b*Log[Cos[(c + d*x)/2]] - 2400*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 1800*a^4*b*Log[Sin[(c + d*x)/2]] + 2400*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 270*a^4*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 656*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 41*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 736*a^5*Tan[(c + d*x)/2] - 1120*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)","A",1
181,1,240,242,6.2548892,"\int \frac{\tan ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[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{7 a+3 b}{16 d (a+b)^3 (1-\sin (c+d x))}-\frac{7 a-3 b}{16 d (a-b)^3 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b)^2 (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b)^2 (\sin (c+d x)+1)^2}-\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{\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{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 ^2(c+d x) \left(2 \left(2 a^4+3 a^2 b^2-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,"-1/8*(a*(4*a + b)*Log[1 - Sin[c + d*x]])/((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) + 1/(16*(a + b)^2*d*(1 - Sin[c + d*x])^2) - (7*a + 3*b)/(16*(a + b)^3*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)^2*d*(1 + Sin[c + d*x])^2) - (7*a - 3*b)/(16*(a - b)^3*d*(1 + Sin[c + d*x])) - a^5/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",1
182,1,145,161,0.7943717,"\int \frac{\tan ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{4 a^2 \left(a^2+3 b^2\right) \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^3}+\frac{4 a^3}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{1}{(a+b)^2 (\sin (c+d x)-1)}+\frac{1}{(a-b)^2 (\sin (c+d x)+1)}+\frac{2 a \log (1-\sin (c+d x))}{(a+b)^3}+\frac{2 a \log (\sin (c+d x)+1)}{(a-b)^3}}{4 d}","-\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^3}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\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,"((2*a*Log[1 - Sin[c + d*x]])/(a + b)^3 + (2*a*Log[1 + Sin[c + d*x]])/(a - b)^3 - (4*a^2*(a^2 + 3*b^2)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^3 - 1/((a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)^2*(1 + Sin[c + d*x])) + (4*a^3)/((a^2 - b^2)^2*(a + b*Sin[c + d*x])))/(4*d)","A",1
183,1,162,109,0.2993376,"\int \frac{\tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + b*Sin[c + d*x])^2,x]","-\frac{a \left(-2 \left(\left(a^2+b^2\right) \log (a+b \sin (c+d x))-a^2+b^2\right)+(a-b)^2 \log (1-\sin (c+d x))+(a+b)^2 \log (\sin (c+d x)+1)\right)+b \sin (c+d x) \left(-2 \left(a^2+b^2\right) \log (a+b \sin (c+d x))+(a-b)^2 \log (1-\sin (c+d x))+(a+b)^2 \log (\sin (c+d x)+1)\right)}{2 d (a-b)^2 (a+b)^2 (a+b \sin (c+d 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}",1,"-1/2*(a*((a - b)^2*Log[1 - Sin[c + d*x]] + (a + b)^2*Log[1 + Sin[c + d*x]] - 2*(-a^2 + b^2 + (a^2 + b^2)*Log[a + b*Sin[c + d*x]])) + b*((a - b)^2*Log[1 - Sin[c + d*x]] + (a + b)^2*Log[1 + Sin[c + d*x]] - 2*(a^2 + b^2)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x])/((a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x]))","A",1
184,1,42,53,0.0814522,"\int \frac{\cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{a}{a+b \sin (c+d x)}-\log (a+b \sin (c+d x))+\log (\sin (c+d x))}{a^2 d}","-\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]] - Log[a + b*Sin[c + d*x]] + a/(a + b*Sin[c + d*x]))/(a^2*d)","A",1
185,1,96,114,0.6305935,"\int \frac{\cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-3 b^2\right) \log (\sin (c+d x))-2 \left(a^2-3 b^2\right) \log (a+b \sin (c+d x))+a^2 \csc ^2(c+d x)+\frac{2 a (a-b) (a+b)}{a+b \sin (c+d x)}-4 a b \csc (c+d x)}{2 a^4 d}","\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}-\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{a^2-b^2}{a^3 d (a+b \sin (c+d x))}",1,"-1/2*(-4*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - 3*b^2)*Log[Sin[c + d*x]] - 2*(a^2 - 3*b^2)*Log[a + b*Sin[c + d*x]] + (2*a*(a - b)*(a + b))/(a + b*Sin[c + d*x]))/(a^4*d)","A",1
186,1,187,188,6.138857,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","-\frac{4 b (a-b) (a+b) \csc (c+d x)}{a^5 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{\left(a^4-6 a^2 b^2+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(a^4-6 a^2 b^2+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{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}+\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}",1,"(-4*(a - b)*b*(a + b)*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",1
187,1,341,333,2.0274767,"\int \frac{\tan ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{12 a^4 b \cos (c+d x)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{24 a^3 \left(a^2+4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}-\frac{4 (4 a+b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 (b-4 a) \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}}{12 d}","\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{a^4 b \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\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{(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,"((24*a^3*(a^2 + 4*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + 1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (4*(4*a + b)*Sin[(c + d*x)/2])/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(-4*a + b)*Sin[(c + d*x)/2])/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (12*a^4*b*Cos[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])))/(12*d)","A",1
188,1,169,200,1.0568196,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{2 a \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)}{\left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","-\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{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{\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*(a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (a^2*b*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
189,1,139,115,0.7644857,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{\frac{4 \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)}{\sqrt{a^2-b^2}}+\frac{2 a b \cos (c+d x)}{a+b \sin (c+d x)}-a \tan \left(\frac{1}{2} (c+d x)\right)+a \cot \left(\frac{1}{2} (c+d x)\right)+4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d}","\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}-\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{\cot (c+d x)}{a d (a+b \sin (c+d x))}",1,"-1/2*((4*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*Cot[(c + d*x)/2] - 4*b*Log[Cos[(c + d*x)/2]] + 4*b*Log[Sin[(c + d*x)/2]] + (2*a*b*Cos[c + d*x])/(a + b*Sin[c + d*x]) - a*Tan[(c + d*x)/2])/(a^3*d)","A",1
190,1,403,238,6.1980048,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\left(3 a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(4 b^3-3 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{2 \left(a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}","\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{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{2 \left(a^4-5 a^2 b^2+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{\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + ((4*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) + ((-3*a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(a^4*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","A",0
191,1,361,424,1.5663738,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","-\frac{1920 \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)+240 b \left(15 a^4-40 a^2 b^2+24 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-240 b \left(15 a^4-40 a^2 b^2+24 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left(196 a^5+1162 a^4 b \sin (c+d x)-562 a^4 b \sin (3 (c+d x))+76 a^4 b \sin (5 (c+d x))-735 a^3 b^2-3060 a^2 b^3 \sin (c+d x)+1470 a^2 b^3 \sin (3 (c+d x))-270 a^2 b^3 \sin (5 (c+d x))-12 \left(16 a^5-85 a^3 b^2+60 a b^4\right) \cos (2 (c+d x))+\left(92 a^5-285 a^3 b^2+180 a b^4\right) \cos (4 (c+d x))+540 a b^4+1800 b^5 \sin (c+d x)-900 b^5 \sin (3 (c+d x))+180 b^5 \sin (5 (c+d x))\right)}{a \csc (c+d x)+b}}{960 a^7 d}","\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 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(15 a^4-82 a^2 b^2+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{b \left(15 a^4-40 a^2 b^2+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(38 a^4-135 a^2 b^2+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{\left(4 a^4-17 a^2 b^2+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(2 a^4-12 a^2 b^2+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{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,"-1/960*(1920*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 240*b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*Log[Cos[(c + d*x)/2]] + 240*b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(196*a^5 - 735*a^3*b^2 + 540*a*b^4 - 12*(16*a^5 - 85*a^3*b^2 + 60*a*b^4)*Cos[2*(c + d*x)] + (92*a^5 - 285*a^3*b^2 + 180*a*b^4)*Cos[4*(c + d*x)] + 1162*a^4*b*Sin[c + d*x] - 3060*a^2*b^3*Sin[c + d*x] + 1800*b^5*Sin[c + d*x] - 562*a^4*b*Sin[3*(c + d*x)] + 1470*a^2*b^3*Sin[3*(c + d*x)] - 900*b^5*Sin[3*(c + d*x)] + 76*a^4*b*Sin[5*(c + d*x)] - 270*a^2*b^3*Sin[5*(c + d*x)] + 180*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x]))/(a^7*d)","A",1
192,1,304,321,6.3413999,"\int \frac{\tan ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","-\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{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(a^4+13 a^2 b^2+10 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{7 a+b}{16 d (a+b)^4 (1-\sin (c+d x))}-\frac{7 a-b}{16 d (a-b)^4 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b)^3 (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b)^3 (\sin (c+d x)+1)^2}","-\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{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(a^4+13 a^2 b^2+10 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{\sec ^2(c+d x) \left(8 a^3 \left(a^2+5 b^2\right)-b \left(27 a^4+22 a^2 b^2-b^4\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}",1,"-1/16*((8*a^2 - 5*a*b - b^2)*Log[1 - Sin[c + d*x]])/((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) + 1/(16*(a + b)^3*d*(1 - Sin[c + d*x])^2) - (7*a + b)/(16*(a + b)^4*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)^3*d*(1 + Sin[c + d*x])^2) - (7*a - b)/(16*(a - b)^4*d*(1 + Sin[c + d*x])) - 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]))","A",1
193,1,196,232,2.2768384,"\int \frac{\tan ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{4 a^2 \left(a^2+3 b^2\right)}{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{4 a \left(a^4+8 a^2 b^2+3 b^4\right) \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^4}+\frac{2 a^3}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{1}{(a+b)^3 (\sin (c+d x)-1)}+\frac{1}{(a-b)^3 (\sin (c+d x)+1)}+\frac{(2 a-b) \log (1-\sin (c+d x))}{(a+b)^4}+\frac{(2 a+b) \log (\sin (c+d x)+1)}{(a-b)^4}}{4 d}","\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{\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{a \left(a^4+8 a^2 b^2+3 b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}+\frac{a^3}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\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]])/(a + b)^4 + ((2*a + b)*Log[1 + Sin[c + d*x]])/(a - b)^4 - (4*a*(a^4 + 8*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 - 1/((a + b)^3*(-1 + Sin[c + d*x])) + 1/((a - b)^3*(1 + Sin[c + d*x])) + (2*a^3)/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) + (4*a^2*(a^2 + 3*b^2))/((a^2 - b^2)^3*(a + b*Sin[c + d*x])))/(4*d)","A",1
194,1,213,149,2.1614048,"\int \frac{\tan (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{2 b}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{4 a b \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^2}+a \left(\frac{b \left(\frac{\left(a^2-b^2\right) \left(-5 a^2-4 a b \sin (c+d x)+b^2\right)}{(a+b \sin (c+d x))^2}+2 \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))\right)}{\left(a^2-b^2\right)^3}+\frac{\log (1-\sin (c+d x))}{(a+b)^3}-\frac{\log (\sin (c+d x)+1)}{(a-b)^3}\right)-\frac{\log (1-\sin (c+d x))}{(a+b)^2}+\frac{\log (\sin (c+d x)+1)}{(a-b)^2}}{2 b d}","-\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]]/(a + b)^2) + Log[1 + Sin[c + d*x]]/(a - b)^2 - (4*a*b*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^2 + (2*b)/((a^2 - b^2)*(a + b*Sin[c + d*x])) + a*(Log[1 - Sin[c + d*x]]/(a + b)^3 - Log[1 + Sin[c + d*x]]/(a - b)^3 + (b*(2*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]] + ((a^2 - b^2)*(-5*a^2 + b^2 - 4*a*b*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/(a^2 - b^2)^3))/(2*b*d)","A",1
195,1,60,75,0.2616917,"\int \frac{\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{a (3 a+2 b \sin (c+d x))}{(a+b \sin (c+d x))^2}-2 \log (a+b \sin (c+d x))+2 \log (\sin (c+d x))}{2 a^3 d}","-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{a^2 d (a+b \sin (c+d x))}+\frac{1}{2 a d (a+b \sin (c+d x))^2}",1,"(2*Log[Sin[c + d*x]] - 2*Log[a + b*Sin[c + d*x]] + (a*(3*a + 2*b*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2)/(2*a^3*d)","A",1
196,1,121,145,0.9398682,"\int \frac{\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","-\frac{\frac{2 a \left(a^2-3 b^2\right)}{a+b \sin (c+d x)}+2 \left(a^2-6 b^2\right) \log (\sin (c+d x))-2 \left(a^2-6 b^2\right) \log (a+b \sin (c+d x))+\frac{a^2 (a-b) (a+b)}{(a+b \sin (c+d x))^2}+a^2 \csc ^2(c+d x)-6 a b \csc (c+d x)}{2 a^5 d}","\frac{3 b \csc (c+d x)}{a^4 d}-\frac{\csc ^2(c+d x)}{2 a^3 d}-\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{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}",1,"-1/2*(-6*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - 6*b^2)*Log[Sin[c + d*x]] - 2*(a^2 - 6*b^2)*Log[a + b*Sin[c + d*x]] + (a^2*(a - b)*(a + b))/(a + b*Sin[c + d*x])^2 + (2*a*(a^2 - 3*b^2))/(a + b*Sin[c + d*x]))/(a^5*d)","A",1
197,1,195,221,5.3057164,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","\frac{-a^4 \csc ^4(c+d x)+\frac{2 \left(a^3-a b^2\right)^2}{(a+b \sin (c+d x))^2}+4 a^3 b \csc ^3(c+d x)+4 a^2 \left(a^2-3 b^2\right) \csc ^2(c+d x)-8 a b \left(3 a^2-5 b^2\right) \csc (c+d x)+\frac{4 a \left(a^4-6 a^2 b^2+5 b^4\right)}{a+b \sin (c+d x)}+4 \left(a^4-12 a^2 b^2+15 b^4\right) \log (\sin (c+d x))-4 \left(a^4-12 a^2 b^2+15 b^4\right) \log (a+b \sin (c+d x))}{4 a^7 d}","\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}-\frac{2 b \left(3 a^2-5 b^2\right) \csc (c+d x)}{a^6 d}+\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{\left(a^4-12 a^2 b^2+15 b^4\right) \log (\sin (c+d x))}{a^7 d}-\frac{\left(a^4-12 a^2 b^2+15 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}+\frac{a^4-6 a^2 b^2+5 b^4}{a^6 d (a+b \sin (c+d x))}",1,"(-8*a*b*(3*a^2 - 5*b^2)*Csc[c + d*x] + 4*a^2*(a^2 - 3*b^2)*Csc[c + d*x]^2 + 4*a^3*b*Csc[c + d*x]^3 - a^4*Csc[c + d*x]^4 + 4*(a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]] - 4*(a^4 - 12*a^2*b^2 + 15*b^4)*Log[a + b*Sin[c + d*x]] + (2*(a^3 - a*b^2)^2)/(a + b*Sin[c + d*x])^2 + (4*a*(a^4 - 6*a^2*b^2 + 5*b^4))/(a + b*Sin[c + d*x]))/(4*a^7*d)","A",1
198,1,351,474,1.0449425,"\int \frac{\tan ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{96 a^2 \left(2 a^4+21 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{9/2}}-\frac{\sec ^3(c+d x) \left(32 a^7 \sin (3 (c+d x))-264 a^6 b+22 a^5 b^2 \sin (c+d x)-91 a^5 b^2 \sin (3 (c+d x))-17 a^5 b^2 \sin (5 (c+d x))-358 a^4 b^3-264 a^3 b^4 \sin (c+d x)-244 a^3 b^4 \sin (3 (c+d x))-76 a^3 b^4 \sin (5 (c+d x))+8 a^2 b^5-2 \left(28 a^6 b+89 a^4 b^3-12 a^2 b^5\right) \cos (4 (c+d x))-8 \left(44 a^6 b+55 a^4 b^3+8 a^2 b^5-2 b^7\right) \cos (2 (c+d x))+32 a b^6 \sin (c+d x)-12 a b^6 \sin (3 (c+d x))-12 a b^6 \sin (5 (c+d x))-16 b^7\right)}{\left(a^2-b^2\right)^4 (a+b \sin (c+d x))^2}}{96 d}","\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 \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{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,"((96*a^2*(2*a^4 + 21*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(9/2) - (Sec[c + d*x]^3*(-264*a^6*b - 358*a^4*b^3 + 8*a^2*b^5 - 16*b^7 - 8*(44*a^6*b + 55*a^4*b^3 + 8*a^2*b^5 - 2*b^7)*Cos[2*(c + d*x)] - 2*(28*a^6*b + 89*a^4*b^3 - 12*a^2*b^5)*Cos[4*(c + d*x)] + 22*a^5*b^2*Sin[c + d*x] - 264*a^3*b^4*Sin[c + d*x] + 32*a*b^6*Sin[c + d*x] + 32*a^7*Sin[3*(c + d*x)] - 91*a^5*b^2*Sin[3*(c + d*x)] - 244*a^3*b^4*Sin[3*(c + d*x)] - 12*a*b^6*Sin[3*(c + d*x)] - 17*a^5*b^2*Sin[5*(c + d*x)] - 76*a^3*b^4*Sin[5*(c + d*x)] - 12*a*b^6*Sin[5*(c + d*x)]))/((a^2 - b^2)^4*(a + b*Sin[c + d*x])^2))/(96*d)","A",1
199,1,212,350,3.2205125,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{2 \left(2 a^4+11 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}-\frac{a b \cos (c+d x) \left(4 a^3+b \left(3 a^2+4 b^2\right) \sin (c+d x)+3 a b^2\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","-\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{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{3 a^3 b \cos (c+d x)}{2 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,"((-2*(2*a^4 + 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (a*b*Cos[c + d*x]*(4*a^3 + 3*a*b^2 + b*(3*a^2 + 4*b^2)*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
200,1,195,202,5.7825197,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{a b \left(4 b^2-3 a^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{(a+b \sin (c+d x))^2}-\frac{2 \left(2 a^4-9 a^2 b^2+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)}{\left(a^2-b^2\right)^{3/2}}+a \tan \left(\frac{1}{2} (c+d x)\right)-a \cot \left(\frac{1}{2} (c+d x)\right)-6 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}","\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\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{\left(2 a^4-9 a^2 b^2+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{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"((-2*(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^2 - b^2)^(3/2) - a*Cot[(c + d*x)/2] + 6*b*Log[Cos[(c + d*x)/2]] - 6*b*Log[Sin[(c + d*x)/2]] - (a^2*b*Cos[c + d*x])/(a + b*Sin[c + d*x])^2 + (a*b*(-3*a^2 + 4*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])) + a*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
201,1,459,289,6.210833,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{3 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{3 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{\left(9 a^2 b-20 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{\left(20 b^3-9 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(2 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-2 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac{\left(2 a^4-19 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}","\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{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 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(2 a^4-19 a^2 b^2+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{\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) + ((2*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*a^5*d) + (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^3*d) + ((-9*a^2*b + 20*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + ((9*a^2*b - 20*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x)/2]^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(3*a^5*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x])^2) + (3*a^2*b*Cos[c + d*x] - 8*b^3*Cos[c + d*x])/(2*a^5*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3*d)","A",0
202,1,448,492,1.7583482,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","\frac{-480 b \left(45 a^4-200 a^2 b^2+168 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 b \left(45 a^4-200 a^2 b^2+168 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{3840 \left(2 a^6-31 a^4 b^2+71 a^2 b^4-42 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 a \cot (c+d x) \csc ^6(c+d x) \left(-784 a^6-8156 a^5 b \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))+3256 a^4 b^2+42270 a^3 b^3 \sin (c+d x)-20715 a^3 b^3 \sin (3 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+7860 a^2 b^4+2 \left(384 a^6-2131 a^4 b^2-6315 a^2 b^4+9450 b^6\right) \cos (2 (c+d x))+\left(-368 a^6+824 a^4 b^2+6060 a^2 b^4-7560 b^6\right) \cos (4 (c+d x))-37800 a b^5 \sin (c+d x)+18900 a b^5 \sin (3 (c+d x))-3780 a b^5 \sin (5 (c+d x))+1260 b^6 \cos (6 (c+d x))-12600 b^6\right)}{(a \csc (c+d x)+b)^2}}{3840 a^8 d}","\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{\left(4 a^4-54 a^2 b^2+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{\sqrt{a^2-b^2} \left(2 a^4-29 a^2 b^2+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{b \left(45 a^4-200 a^2 b^2+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(91 a^4-645 a^2 b^2+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{\left(8 a^4-79 a^2 b^2+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left(15 a^4-187 a^2 b^2+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(5 a^4-60 a^2 b^2+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{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,"((-3840*(2*a^6 - 31*a^4*b^2 + 71*a^2*b^4 - 42*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Cos[(c + d*x)/2]] - 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(-784*a^6 + 3256*a^4*b^2 + 7860*a^2*b^4 - 12600*b^6 + 2*(384*a^6 - 2131*a^4*b^2 - 6315*a^2*b^4 + 9450*b^6)*Cos[2*(c + d*x)] + (-368*a^6 + 824*a^4*b^2 + 6060*a^2*b^4 - 7560*b^6)*Cos[4*(c + d*x)] + 182*a^4*b^2*Cos[6*(c + d*x)] - 1290*a^2*b^4*Cos[6*(c + d*x)] + 1260*b^6*Cos[6*(c + d*x)] - 8156*a^5*b*Sin[c + d*x] + 42270*a^3*b^3*Sin[c + d*x] - 37800*a*b^5*Sin[c + d*x] + 3956*a^5*b*Sin[3*(c + d*x)] - 20715*a^3*b^3*Sin[3*(c + d*x)] + 18900*a*b^5*Sin[3*(c + d*x)] - 608*a^5*b*Sin[5*(c + d*x)] + 3975*a^3*b^3*Sin[5*(c + d*x)] - 3780*a*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(3840*a^8*d)","A",1
203,1,4791,271,16.0955768,"\int (a+b \sin (e+f x))^3 (g \tan (e+f x))^p \, dx","Integrate[(a + b*Sin[e + f*x])^3*(g*Tan[e + f*x])^p,x]","\text{Result too large to show}","\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^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{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,"(2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Tan[(e + f*x)/2]*(a^3*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + p)*(3*a^2*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b^2*(AppellF1[1 + p/2, p, 3, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + p/2, p, 4, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*Tan[(e + f*x)/2]))*(g*Tan[e + f*x])^p*(-1/8*(b^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p) - a^3*Sin[e + f*x]^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p + ((3*I)/8)*b^3*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p + (3*b^3*Sin[2*(e + f*x)]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/8 - (I/8)*b^3*Sin[2*(e + f*x)]^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p + Cos[e + f*x]^3*(a^3*Cos[3*(e + f*x)]*Tan[e + f*x]^p - I*a^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p) + Cos[2*(e + f*x)]^3*((I/8)*b^3*Cos[3*(e + f*x)]*Tan[e + f*x]^p + (b^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/8) + Sin[e + f*x]^2*((-3*a^2*b*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 + ((3*I)/2)*a^2*b*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p) + Sin[e + f*x]*((-3*a*b^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/4 + ((3*I)/2)*a*b^2*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p + (3*a*b^2*Sin[2*(e + f*x)]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/4) + Cos[2*(e + f*x)]^2*((-3*b^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/8 - (3*a*b^2*Sin[e + f*x]*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/4 + ((3*I)/8)*b^3*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p + Cos[3*(e + f*x)]*(((-3*I)/8)*b^3*Tan[e + f*x]^p - ((3*I)/4)*a*b^2*Sin[e + f*x]*Tan[e + f*x]^p - (3*b^3*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/8)) + Cos[3*(e + f*x)]*((-1/8*I)*b^3*Tan[e + f*x]^p - I*a^3*Sin[e + f*x]^3*Tan[e + f*x]^p - (3*b^3*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/8 + ((3*I)/8)*b^3*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p + (b^3*Sin[2*(e + f*x)]^3*Tan[e + f*x]^p)/8 + Sin[e + f*x]^2*(((-3*I)/2)*a^2*b*Tan[e + f*x]^p - (3*a^2*b*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/2) + Sin[e + f*x]*(((-3*I)/4)*a*b^2*Tan[e + f*x]^p - (3*a*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/2 + ((3*I)/4)*a*b^2*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p)) + Cos[e + f*x]^2*((3*a^2*b*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 + 3*a^3*Sin[e + f*x]*Sin[3*(e + f*x)]*Tan[e + f*x]^p - ((3*I)/2)*a^2*b*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p + Cos[3*(e + f*x)]*(((3*I)/2)*a^2*b*Tan[e + f*x]^p + (3*I)*a^3*Sin[e + f*x]*Tan[e + f*x]^p + (3*a^2*b*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/2) + Cos[2*(e + f*x)]*(((-3*I)/2)*a^2*b*Cos[3*(e + f*x)]*Tan[e + f*x]^p - (3*a^2*b*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2)) + Cos[e + f*x]*(((3*I)/4)*a*b^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p + (3*I)*a^3*Sin[e + f*x]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p + (3*a*b^2*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 - ((3*I)/4)*a*b^2*Sin[2*(e + f*x)]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p + Cos[2*(e + f*x)]^2*((-3*a*b^2*Cos[3*(e + f*x)]*Tan[e + f*x]^p)/4 + ((3*I)/4)*a*b^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p) + Sin[e + f*x]*((3*I)*a^2*b*Sin[3*(e + f*x)]*Tan[e + f*x]^p + 3*a^2*b*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p) + Cos[3*(e + f*x)]*((-3*a*b^2*Tan[e + f*x]^p)/4 - 3*a^3*Sin[e + f*x]^2*Tan[e + f*x]^p + ((3*I)/2)*a*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p + (3*a*b^2*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p)/4 + Sin[e + f*x]*(-3*a^2*b*Tan[e + f*x]^p + (3*I)*a^2*b*Sin[2*(e + f*x)]*Tan[e + f*x]^p)) + Cos[2*(e + f*x)]*(((-3*I)/2)*a*b^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p - (3*I)*a^2*b*Sin[e + f*x]*Sin[3*(e + f*x)]*Tan[e + f*x]^p - (3*a*b^2*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 + Cos[3*(e + f*x)]*((3*a*b^2*Tan[e + f*x]^p)/2 + 3*a^2*b*Sin[e + f*x]*Tan[e + f*x]^p - ((3*I)/2)*a*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p))) + Cos[2*(e + f*x)]*((3*b^3*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/8 + (3*a^2*b*Sin[e + f*x]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 - ((3*I)/4)*b^3*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p - (3*b^3*Sin[2*(e + f*x)]^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/8 + Sin[e + f*x]*((3*a*b^2*Sin[3*(e + f*x)]*Tan[e + f*x]^p)/2 - ((3*I)/2)*a*b^2*Sin[2*(e + f*x)]*Sin[3*(e + f*x)]*Tan[e + f*x]^p) + Cos[3*(e + f*x)]*(((3*I)/8)*b^3*Tan[e + f*x]^p + ((3*I)/2)*a^2*b*Sin[e + f*x]^2*Tan[e + f*x]^p + (3*b^3*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/4 - ((3*I)/8)*b^3*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p + Sin[e + f*x]*(((3*I)/2)*a*b^2*Tan[e + f*x]^p + (3*a*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p)/2)))))/(f*(1 + p)*(2 + p)*((2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Sec[e + f*x]^2*Tan[(e + f*x)/2]*(a^3*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + p)*(3*a^2*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b^2*(AppellF1[1 + p/2, p, 3, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + p/2, p, 4, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*Tan[(e + f*x)/2]))*Tan[e + f*x]^(-1 + p))/((1 + p)*(2 + p)) + (Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*(a^3*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + p)*(3*a^2*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b^2*(AppellF1[1 + p/2, p, 3, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + p/2, p, 4, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*Tan[(e + f*x)/2]))*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + p)*Tan[(e + f*x)/2]*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*(a^3*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 6*a*b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + p)*(3*a^2*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b^2*(AppellF1[1 + p/2, p, 3, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + p/2, p, 4, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*Tan[(e + f*x)/2]))*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Tan[(e + f*x)/2]*(a^3*(2 + p)*(-(((1 + p)*AppellF1[1 + (1 + p)/2, p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 1, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + 2*b*(((1 + p)*(3*a^2*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b^2*(AppellF1[1 + p/2, p, 3, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - AppellF1[1 + p/2, p, 4, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]))*Sec[(e + f*x)/2]^2)/2 + 6*a*b*(2 + p)*((-2*(1 + p)*AppellF1[1 + (1 + p)/2, p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) - 6*a*b*(2 + p)*((-3*(1 + p)*AppellF1[1 + (1 + p)/2, p, 4, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + (1 + p)*Tan[(e + f*x)/2]*(3*a^2*((-2*(1 + p/2)*AppellF1[2 + p/2, p, 3, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) + ((1 + p/2)*p*AppellF1[2 + p/2, 1 + p, 2, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2)) + 4*b^2*((-3*(1 + p/2)*AppellF1[2 + p/2, p, 4, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) + (4*(1 + p/2)*AppellF1[2 + p/2, p, 5, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) + ((1 + p/2)*p*AppellF1[2 + p/2, 1 + p, 3, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) - ((1 + p/2)*p*AppellF1[2 + p/2, 1 + p, 4, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2)))))*Tan[e + f*x]^p)/((1 + p)*(2 + p))))","C",0
204,1,2464,186,14.2359086,"\int (a+b \sin (e+f x))^2 (g \tan (e+f x))^p \, dx","Integrate[(a + b*Sin[e + f*x])^2*(g*Tan[e + f*x])^p,x]","\text{Result too large to show}","\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,"(2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Tan[(e + f*x)/2]*(a^2*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b*(b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + a*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]))*(g*Tan[e + f*x])^p*(-1/4*(b^2*Cos[2*(e + f*x)]^3*Tan[e + f*x]^p) + (I/4)*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p + I*a^2*Sin[e + f*x]^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p + (b^2*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p)/2 - (I/4)*b^2*Sin[2*(e + f*x)]^3*Tan[e + f*x]^p + Cos[e + f*x]^2*(a^2*Cos[2*(e + f*x)]*Tan[e + f*x]^p - I*a^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p) + Cos[2*(e + f*x)]^2*((b^2*Tan[e + f*x]^p)/2 + a*b*Sin[e + f*x]*Tan[e + f*x]^p - (I/4)*b^2*Sin[2*(e + f*x)]*Tan[e + f*x]^p) + Sin[e + f*x]*(I*a*b*Sin[2*(e + f*x)]*Tan[e + f*x]^p + a*b*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p) + Cos[2*(e + f*x)]*(-1/4*(b^2*Tan[e + f*x]^p) - a*b*Sin[e + f*x]*Tan[e + f*x]^p - a^2*Sin[e + f*x]^2*Tan[e + f*x]^p - (b^2*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p)/4) + Cos[e + f*x]*((-I)*a*b*Cos[2*(e + f*x)]^2*Tan[e + f*x]^p + a*b*Sin[2*(e + f*x)]*Tan[e + f*x]^p + 2*a^2*Sin[e + f*x]*Sin[2*(e + f*x)]*Tan[e + f*x]^p - I*a*b*Sin[2*(e + f*x)]^2*Tan[e + f*x]^p + Cos[2*(e + f*x)]*(I*a*b*Tan[e + f*x]^p + (2*I)*a^2*Sin[e + f*x]*Tan[e + f*x]^p))))/(f*(1 + p)*(2 + p)*((2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Sec[e + f*x]^2*Tan[(e + f*x)/2]*(a^2*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b*(b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + a*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]))*Tan[e + f*x]^(-1 + p))/((1 + p)*(2 + p)) + (Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*(a^2*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b*(b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + a*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]))*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*p*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + p)*Tan[(e + f*x)/2]*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*(a^2*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*b*(b*(2 + p)*AppellF1[(1 + p)/2, p, 2, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*(2 + p)*AppellF1[(1 + p)/2, p, 3, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + a*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]))*Tan[e + f*x]^p)/((1 + p)*(2 + p)) + (2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^p*Tan[(e + f*x)/2]*(a^2*(2 + p)*(-(((1 + p)*AppellF1[1 + (1 + p)/2, p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 1, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) + 4*b*((a*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/2 + a*(1 + p)*Tan[(e + f*x)/2]*((-2*(1 + p/2)*AppellF1[2 + p/2, p, 3, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2) + ((1 + p/2)*p*AppellF1[2 + p/2, 1 + p, 2, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + p/2)) + b*(2 + p)*((-2*(1 + p)*AppellF1[1 + (1 + p)/2, p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 2, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p)) - b*(2 + p)*((-3*(1 + p)*AppellF1[1 + (1 + p)/2, p, 4, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p) + (p*(1 + p)*AppellF1[1 + (1 + p)/2, 1 + p, 3, 1 + (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3 + p))))*Tan[e + f*x]^p)/((1 + p)*(2 + p))))","C",0
205,1,849,129,9.533954,"\int (a+b \sin (e+f x)) (g \tan (e+f x))^p \, dx","Integrate[(a + b*Sin[e + f*x])*(g*Tan[e + f*x])^p,x]","\frac{2 (a+b \sin (e+f x)) \tan \left(\frac{1}{2} (e+f x)\right) \left(a (p+2) F_1\left(\frac{p+1}{2};p,1;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 b (p+1) F_1\left(\frac{p}{2}+1;p,2;\frac{p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)\right) (g \tan (e+f x))^p}{f \left(-16 p \cos \left(\frac{1}{2} (e+f x)\right) \csc ^3(e+f x) \sec (e+f x) \left(a (p+2) F_1\left(\frac{p+1}{2};p,1;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 b (p+1) F_1\left(\frac{p}{2}+1;p,2;\frac{p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)\right) \sin ^5\left(\frac{1}{2} (e+f x)\right)+\sec ^2\left(\frac{1}{2} (e+f x)\right) \left(a (p+2) F_1\left(\frac{p+1}{2};p,1;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 b (p+1) F_1\left(\frac{p}{2}+1;p,2;\frac{p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)\right)+2 p \csc (e+f x) \sec (e+f x) \tan \left(\frac{1}{2} (e+f x)\right) \left(a (p+2) F_1\left(\frac{p+1}{2};p,1;\frac{p+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+2 b (p+1) F_1\left(\frac{p}{2}+1;p,2;\frac{p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)\right)+2 (p+1) \sec ^2\left(\frac{1}{2} (e+f x)\right) \tan \left(\frac{1}{2} (e+f x)\right) \left(\frac{2 b (p+2) \left(p F_1\left(\frac{p}{2}+2;p+1,2;\frac{p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 F_1\left(\frac{p}{2}+2;p,3;\frac{p}{2}+3;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (e+f x)\right)}{p+4}+\frac{a (p+2) \left(p F_1\left(\frac{p+3}{2};p+1,1;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-F_1\left(\frac{p+3}{2};p,2;\frac{p+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right) \tan \left(\frac{1}{2} (e+f x)\right)}{p+3}+b F_1\left(\frac{p}{2}+1;p,2;\frac{p}{2}+2;\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","\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,"(2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2]*(a*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])*(g*Tan[e + f*x])^p)/(f*(Sec[(e + f*x)/2]^2*(a*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]) - 16*p*Cos[(e + f*x)/2]*Csc[e + f*x]^3*Sec[e + f*x]*Sin[(e + f*x)/2]^5*(a*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]) + 2*p*Csc[e + f*x]*Sec[e + f*x]*Tan[(e + f*x)/2]*(a*(2 + p)*AppellF1[(1 + p)/2, p, 1, (3 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*b*(1 + p)*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]) + 2*(1 + p)*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*(b*AppellF1[1 + p/2, p, 2, 2 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (a*(2 + p)*(-AppellF1[(3 + p)/2, p, 2, (5 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[(3 + p)/2, 1 + p, 1, (5 + p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2])/(3 + p) + (2*b*(2 + p)*(-2*AppellF1[2 + p/2, p, 3, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + p*AppellF1[2 + p/2, 1 + p, 2, 3 + p/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/(4 + p))))","C",0
206,1,858,284,13.5896933,"\int \frac{(g \tan (e+f x))^p}{a+b \sin (e+f x)} \, dx","Integrate[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x]),x]","\frac{\tan ^{p+1}(e+f x) (g \tan (e+f x))^p \left(\left(a^2-b^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},1;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (p+2) \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (p+1) \, _2F_1\left(\frac{1}{2},\frac{p}{2}+1;\frac{p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right)}{a^2 b f (p+1) (p+2) (a+b \sin (e+f x)) \left(\frac{\sec ^2(e+f x) \left(\left(a^2-b^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},1;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)+a \left(b (p+2) \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)-a (p+1) \, _2F_1\left(\frac{1}{2},\frac{p}{2}+1;\frac{p}{2}+2;-\tan ^2(e+f x)\right) \tan (e+f x)\right)\right) \tan ^p(e+f x)}{a^2 b (p+2)}+\frac{\left(\left(a^2-b^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},1;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+\left(a^2-b^2\right) (p+1) \tan (e+f x) \left(\frac{2 \left(\frac{b^2}{a^2}-1\right) (p+2) F_1\left(\frac{p+2}{2}+1;-\frac{1}{2},2;\frac{p+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{p+4}+\frac{(p+2) F_1\left(\frac{p+2}{2}+1;\frac{1}{2},1;\frac{p+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{p+4}\right)+a \left(-a (p+1) \, _2F_1\left(\frac{1}{2},\frac{p}{2}+1;\frac{p}{2}+2;-\tan ^2(e+f x)\right) \sec ^2(e+f x)-2 a \left(\frac{p}{2}+1\right) (p+1) \left(\frac{1}{\sqrt{\tan ^2(e+f x)+1}}-\, _2F_1\left(\frac{1}{2},\frac{p}{2}+1;\frac{p}{2}+2;-\tan ^2(e+f x)\right)\right) \sec ^2(e+f x)+b (p+1) (p+2) \csc (e+f x) \left(\frac{1}{1-\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}}-\, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right)\right) \sec (e+f x)\right)\right) \tan ^{p+1}(e+f x)}{a^2 b (p+1) (p+2)}\right)}","\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,"(Tan[e + f*x]^(1 + p)*(g*Tan[e + f*x])^p*((a^2 - b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 1, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + p)*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + p)*Hypergeometric2F1[1/2, 1 + p/2, 2 + p/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*f*(1 + p)*(2 + p)*(a + b*Sin[e + f*x])*((Sec[e + f*x]^2*Tan[e + f*x]^p*((a^2 - b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 1, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x] + a*(b*(2 + p)*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] - a*(1 + p)*Hypergeometric2F1[1/2, 1 + p/2, 2 + p/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a^2*b*(2 + p)) + (Tan[e + f*x]^(1 + p)*((a^2 - b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 1, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + (a^2 - b^2)*(1 + p)*Tan[e + f*x]*((2*(-1 + b^2/a^2)*(2 + p)*AppellF1[1 + (2 + p)/2, -1/2, 2, 1 + (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + p) + ((2 + p)*AppellF1[1 + (2 + p)/2, 1/2, 1, 1 + (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + p)) + a*(-(a*(1 + p)*Hypergeometric2F1[1/2, 1 + p/2, 2 + p/2, -Tan[e + f*x]^2]*Sec[e + f*x]^2) - 2*a*(1 + p/2)*(1 + p)*Sec[e + f*x]^2*(-Hypergeometric2F1[1/2, 1 + p/2, 2 + p/2, -Tan[e + f*x]^2] + 1/Sqrt[1 + Tan[e + f*x]^2]) + b*(1 + p)*(2 + p)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2] + (1 - ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2)^(-1)))))/(a^2*b*(1 + p)*(2 + p))))","B",0
207,1,866,737,14.3968803,"\int \frac{(g \tan (e+f x))^p}{(a+b \sin (e+f x))^2} \, dx","Integrate[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^2,x]","\frac{\tan ^{p+1}(e+f x) (g \tan (e+f x))^p \left(a (p+2) \left(\left(a^2+b^2\right) \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 \, _2F_1\left(2,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)+2 b \left(b^2-a^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},2;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right)}{a^3 \left(a^2-b^2\right) f (p+1) (p+2) (a+b \sin (e+f x))^2 \left(\frac{\sec ^2(e+f x) \left(a (p+2) \left(\left(a^2+b^2\right) \, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)-2 b^2 \, _2F_1\left(2,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)+2 b \left(b^2-a^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},2;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x)\right) \tan ^p(e+f x)}{a^3 \left(a^2-b^2\right) (p+2)}+\frac{\left(2 b \left(b^2-a^2\right) (p+1) F_1\left(\frac{p+2}{2};-\frac{1}{2},2;\frac{p+4}{2};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sec ^2(e+f x)+2 b \left(b^2-a^2\right) (p+1) \tan (e+f x) \left(\frac{4 \left(\frac{b^2}{a^2}-1\right) (p+2) F_1\left(\frac{p+2}{2}+1;-\frac{1}{2},3;\frac{p+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{p+4}+\frac{(p+2) F_1\left(\frac{p+2}{2}+1;\frac{1}{2},2;\frac{p+4}{2}+1;-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan (e+f x) \sec ^2(e+f x)}{p+4}\right)+a (p+2) \left(\left(a^2+b^2\right) (p+1) \csc (e+f x) \sec (e+f x) \left(\frac{1}{1-\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)}-\, _2F_1\left(1,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)-2 b^2 (p+1) \csc (e+f x) \sec (e+f x) \left(\frac{1}{\left(1-\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)^2}-\, _2F_1\left(2,\frac{p+1}{2};\frac{p+3}{2};\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right)\right)\right) \tan ^{p+1}(e+f x)}{a^3 \left(a^2-b^2\right) (p+1) (p+2)}\right)}","-\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,"(Tan[e + f*x]^(1 + p)*(g*Tan[e + f*x])^p*(a*(2 + p)*((a^2 + b^2)*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*Hypergeometric2F1[2, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(-a^2 + b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 2, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^3*(a^2 - b^2)*f*(1 + p)*(2 + p)*(a + b*Sin[e + f*x])^2*((Sec[e + f*x]^2*Tan[e + f*x]^p*(a*(2 + p)*((a^2 + b^2)*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2] - 2*b^2*Hypergeometric2F1[2, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(-a^2 + b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 2, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]))/(a^3*(a^2 - b^2)*(2 + p)) + (Tan[e + f*x]^(1 + p)*(2*b*(-a^2 + b^2)*(1 + p)*AppellF1[(2 + p)/2, -1/2, 2, (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + 2*b*(-a^2 + b^2)*(1 + p)*Tan[e + f*x]*((4*(-1 + b^2/a^2)*(2 + p)*AppellF1[1 + (2 + p)/2, -1/2, 3, 1 + (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + p) + ((2 + p)*AppellF1[1 + (2 + p)/2, 1/2, 2, 1 + (4 + p)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(4 + p)) + a*(2 + p)*(-2*b^2*(1 + p)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[2, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 - (-1 + b^2/a^2)*Tan[e + f*x]^2)^(-2)) + (a^2 + b^2)*(1 + p)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (1 - (-1 + b^2/a^2)*Tan[e + f*x]^2)^(-1)))))/(a^3*(a^2 - b^2)*(1 + p)*(2 + p))))","A",0
208,0,0,26,3.1289396,"\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx","Integrate[(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,"Integrate[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x]","A",-1